Reduction of Fifth-Order Ordinary Differential Equations to Linearizable form by Contact Transformations

Abstract

This paper is devoted to the study of the linearization problem of fifth-order ordinary differential equations by means of contact transformations. The necessary and sufficient conditions for linearization are obtained. The procedure for obtaining the linearizing transformations is provided in explicit form. Examples demonstrating the procedure of using the linearization theorems are presented.

Appendix

For proving the theorems one needs relations between $\varphi (x,y,p), \psi (x,y,p), g(x,y,p)$ and coefficients of equation (3.5). These relations are presented here.

$$\begin{aligned} A_2&= ((4 (g_{py} p - g_{pp} r + g_{px}) - (4 g_{p} r_{p} - 3 g_{y})) \varphi _{p} - (6 \varphi _{pp} (g_{x} + g_{y} p - g_{p} r)\nonumber \\&\quad - \varphi _{y} g_{p}))/((g_{x} + g_{y} p - g_{p} r) \varphi _{p}), \end{aligned}$$

(8.1)

$$\begin{aligned} A_1&= - (2 (2 (4 g_{x} r_{p} - g_{yy} p^2 - 2 g_{xy} p + g_{pp} r^2) + (8 r_{p} p - r) g_{y} - 2 g_{xx}\nonumber \\&\quad + 2 (r_{x} + r_{y} p - 3 r_{p} r) g_{p})\varphi _{p} - ((7 (g_{x} + g_{y} p) - 5 g_{p} r) \varphi _{y}\nonumber \\&\quad - 12 (g_{x} + g_{y} p - g_{p} r) \varphi _{pp} r)))/((g_{x} + g_{y} p - g_{p} r)\varphi _{p}), \end{aligned}$$

(8.2)

$$\begin{aligned} A_0&= (2 p^2 \varphi _{p} (2 g_{yy} r - 5 g_{y} r_{y}) + p ( - 6 \varphi _{pp} g_{y} r^2 - 4 \varphi _{p} g_{py} r^2+ 6 \varphi _{p} g_{p} r_{y} r\nonumber \\&\quad + 8 \varphi _{p} g_{xy} r - 10 \varphi _{p} g_{x} r_{y} - 6 \varphi _{p} g_{y} r_{p} r - 10 \varphi _{p} g_{y} r_{x} + 7 \varphi _{y} g_{y} r)\nonumber \\&\quad + 6 \varphi _{pp} g_{p} r^3 - 6 \varphi _{pp} g_{x} r^2 - 4 \varphi _{p} g_{px} r^2 + 6 \varphi _{p} g_{p} r_{p} r^2 + 6 \varphi _{p} g_{p} r_{x} r \nonumber \\&\quad + 4 \varphi _{p} g_{xx} r - 6 \varphi _{p} g_{x} r_{p} r - 10 \varphi _{p} g_{x} r_{x} - \varphi _{p} g_{y} r^2 - 6 \varphi _{y} g_{p} r^2 \nonumber \\&\quad + 7 \varphi _{y} g_{x} r)/((g_{x} + g_{y} p - g_{p} r) \varphi _{p}), \end{aligned}$$

(8.3)

$$\begin{aligned} B_2&= 3 ((4 g_{p} r_{p} - 3 g_{y} + 4 g_{pp} r - 4 g_{py} p- 4 g_{px}) \varphi _{p} + 6 (g_{x} + g_{y} p - g_{p} r) \varphi _{pp}\nonumber \\&\quad -\varphi _{y} g_{p})/((g_{x} + g_{y} p - g_{p} r) \varphi _{p}), \end{aligned}$$

(8.4)

$$\begin{aligned} B_1&= - (36 \varphi _{pp} g_{p} r^2 - 36 \varphi _{pp} g_{x} r - 36 \varphi _{pp} g_{y} p r - 6 \varphi _{p} g_{px} r - 6 \varphi _{p} g_{py} p r\nonumber \\&\quad -9 \varphi _{p} g_{pp} r^2 + 51 \varphi _{p} g_{p} r_{p} r - 15 \varphi _{p} g_{p} r_{x} - 15 \varphi _{p} g_{p} r_{y} p + 30 \varphi _{p} g_{xy} p\nonumber \\&\quad + 15 \varphi _{p} g_{xx} - 60 \varphi _{p} g_{x} r_{p} + 15 \varphi _{p} g_{yy} p^2 - 60 \varphi _{p} g_{y} r_{p} p + 3 \varphi _{p} g_{y} r \nonumber \\&\quad - 19 \varphi _{y} g_{p} r + 25 \varphi _{y} g_{x} + 25 \varphi _{y} g_{y} p)/((g_{x} + g_{y} p - g_{p} r) \varphi _{p}), \end{aligned}$$

(8.5)

$$\begin{aligned} B_0&= - (18 \varphi _{pp} g_{p} r^3 - 18 \varphi _{pp} g_{x} r^2 - 18 \varphi _{pp} g_{y} p r^2 - 18 \varphi _{p} g_{px} r^2 - 18 \varphi _{p} g_{py} p r^2\nonumber \\&\quad + 3 \varphi _{p} g_{pp} r^3 + 18 \varphi _{p} g_{p} r_{p} r^2 + 30 \varphi _{p} g_{p} r_{x} r + 30 \varphi _{p} g_{p} r_{y} p r + 30 \varphi _{p} g_{xy} p r\nonumber \\&\quad + 15 \varphi _{p} g_{xx} r - 15 \varphi _{p} g_{x} r_{p} r - 45 \varphi _{p} g_{x} r_{x} - 45 \varphi _{p} g_{x} r_{y} p + 15 \varphi _{p} g_{yy} p^2 r\nonumber \\&\quad - 15 \varphi _{p} g_{y} r_{p} p r - 45 \varphi _{p} g_{y} r_{x} p - 45 \varphi _{p} g_{y} r_{y} p^2 - 6 \varphi _{p} g_{y} r^2 - 22 \varphi _{y} g_{p} r^2 \nonumber \\&\quad + 25 \varphi _{y} g_{x} r + 25 \varphi _{y} g_{y} p r)/((g_{x} + g_{y} p - g_{p} r) \varphi _{p}), \end{aligned}$$

(8.6)

$$\begin{aligned} C_4&=(p ( - 4 \varphi _{ppp} \varphi _{p} g_{y} + 15 \varphi _{pp}^2 g_{y} - 18 \varphi _{pp} \varphi _{p} g_{py} + 6 \varphi _{p}^2 g_{ppy}) + 4 \varphi _{py} \varphi _{p} g_{p}\nonumber \\&\quad + 4 \varphi _{ppp} \varphi _{p} g_{p} r - 4 \varphi _{ppp} \varphi _{p} g_{x} - 15 \varphi _{pp}^2 g_{p} r + 15 \varphi _{pp}^2 g_{x} - 18 \varphi _{pp} \varphi _{p} g_{px}\nonumber \\&\quad + 18 \varphi _{pp} \varphi _{p} g_{pp} r + 18 \varphi _{pp} \varphi _{p} g_{p} r_{p} - 12 \varphi _{pp} \varphi _{p} g_{y} - 10 \varphi _{pp} \varphi _{y} g_{p} + 8 \varphi _{p}^2 g_{py}\nonumber \\&\quad - 6 \varphi _{p}^2 g_{ppp} r + 6 \varphi _{p}^2 g_{ppx} - 12 \varphi _{p}^2 g_{pp} r_{p} - 6 \varphi _{p}^2 g_{p} r_{pp} + 4 \varphi _{p} \varphi _{y} g_{pp})/\nonumber \\&\quad \times ((g_{x} + g_{y} p - g_{p} r) \varphi _{p}^2), \end{aligned}$$

(8.7)

$$\begin{aligned} C_3&=2 (8 \varphi _{py} \varphi _{p} g_{x} + 8 \varphi _{py} \varphi _{p} g_{y} p + 8 \varphi _{ppp} \varphi _{p} g_{p} r^2 - 8 \varphi _{ppp} \varphi _{p} g_{x} r -8 \varphi _{ppp} \varphi _{p} g_{y} p r\nonumber \\&\quad - 30 \varphi _{pp}^2 g_{p} r^2 + 30 \varphi _{pp}^2 g_{x} r + 30 \varphi _{pp}^2 g_{y} p r - 18 \varphi _{pp} \varphi _{p} g_{px} r - 18 \varphi _{pp} \varphi _{p} g_{py} p r\nonumber \\&\quad + 27 \varphi _{pp} \varphi _{p} g_{pp} r^2 + 9 \varphi _{pp} \varphi _{p} g_{p} r_{x} + 9 \varphi _{pp} \varphi _{p} g_{p} r_{y} p - 18 \varphi _{pp} \varphi _{p} g_{xy} p - 9 \varphi _{pp} \varphi _{p} g_{xx}\nonumber \\&\quad + 27 \varphi _{pp} \varphi _{p} g_{x} r_{p} - 9 \varphi _{pp} \varphi _{p} g_{yy} p^2 + 27 \varphi _{pp} \varphi _{p} g_{y} r_{p} p - 15 \varphi _{pp} \varphi _{p} g_{y} r - 20 \varphi _{pp} \varphi _{y} g_{x}\nonumber \\&\quad - 20 \varphi _{pp} \varphi _{y} g_{y} p + 12 \varphi _{p}^2 g_{pxy} p + 6 \varphi _{p}^2 g_{pxx} - 30 \varphi _{p}^2 g_{px} r_{p} + 6 \varphi _{p}^2 g_{pyy} p^2 - 30 \varphi _{p}^2 g_{py} r_{p} p\nonumber \\&\quad + 3 \varphi _{p}^2 g_{py} r - 6 \varphi _{p}^2 g_{ppp} r^2+ 12 \varphi _{p}^2 g_{pp} r_{p} r - 6 \varphi _{p}^2 g_{pp} r_{x} - 6 \varphi _{p}^2 g_{pp} r_{y} p - 6 \varphi _{p}^2 g_{p} r_{px}\nonumber \\&\quad - 6 \varphi _{p}^2 g_{p} r_{py} p + 3 \varphi _{p}^2 g_{p} r_{pp} r + 24 \varphi _{p}^2 g_{p} r_{p}^2 - \varphi _{p}^2 g_{p} r_{y} + 7 \varphi _{p}^2 g_{xy} - 9 \varphi _{p}^2 g_{x} r_{pp}\nonumber \\&\quad + 7 \varphi _{p}^2 g_{yy} p - 9 \varphi _{p}^2 g_{y} r_{pp} p - 18 \varphi _{p}^2 g_{y} r_{p} + 13 \varphi _{p} \varphi _{y} g_{px} + 13 \varphi _{p} \varphi _{y} g_{py} p \nonumber \\&\quad - 5 \varphi _{p} \varphi _{y} g_{pp} r - 19 \varphi _{p} \varphi _{y} g_{p} r_{p} + 8 \varphi _{p} \varphi _{y} g_{y} + 5 \varphi _{y}^2 g_{p})/((g_{x} + g_{y} p - g_{p} r) \varphi _{p}^2), \end{aligned}$$

(8.8)

$$\begin{aligned} C_2&= - (24 \varphi _{py} \varphi _{p} g_{p} r^2 - 48 \varphi _{py} \varphi _{p} g_{x} r - 48 \varphi _{py} \varphi _{p} g_{y} p r - 24 \varphi _{ppp} \varphi _{p} g_{p} r^3 + 24 \varphi _{ppp} \varphi _{p} g_{x} r^2\nonumber \\&\quad + 24 \varphi _{ppp} \varphi _{p} g_{y} p r^2 + 90 \varphi _{pp}^2 g_{p} r^3 - 90 \varphi _{pp}^2 g_{x} r^2 - 90 \varphi _{pp}^2 g_{y} p r^2 - 54 \varphi _{pp} \varphi _{p} g_{pp} r^3\nonumber \\&\quad + 72 \varphi _{pp} \varphi _{p} g_{p} r_{p} r^2 - 18 \varphi _{pp} \varphi _{p} g_{p} r_{x} r - 18 \varphi _{pp} \varphi _{p} g_{p} r_{y} p r + 108 \varphi _{pp} \varphi _{p} g_{xy} p r\nonumber \\&\quad + 54 \varphi _{pp} \varphi _{p} g_{xx} r - 126 \varphi _{pp} \varphi _{p} g_{x} r_{p} r - 36 \varphi _{pp} \varphi _{p} g_{x} r_{x} - 36 \varphi _{pp} \varphi _{p} g_{x} r_{y} p\nonumber \\&\quad + 54 \varphi _{pp} \varphi _{p} g_{yy} p^2 r - 126 \varphi _{pp} \varphi _{p} g_{y} r_{p} p r - 36 \varphi _{pp} \varphi _{p} g_{y} r_{x} p - 36 \varphi _{pp} \varphi _{p} g_{y} r_{y} p^2\nonumber \\&\quad + 18 \varphi _{pp} \varphi _{p} g_{y} r^2 - 60 \varphi _{pp} \varphi _{y} g_{p} r^2 + 120 \varphi _{pp} \varphi _{y} g_{x} r + 120 \varphi _{pp} \varphi _{y} g_{y} p r\nonumber \\&\quad - 36 \varphi _{p}^2 g_{pxy} p r - 18 \varphi _{p}^2 g_{pxx} r + 42 \varphi _{p}^2 g_{px} r_{p} r + 42 \varphi _{p}^2 g_{px} r_{x} + 42 \varphi _{p}^2 g_{px} r_{y} p\nonumber \\&\quad - 18 \varphi _{p}^2 g_{pyy} p^2 r + 42 \varphi _{p}^2 g_{py} r_{p} p r + 42 \varphi _{p}^2 g_{py} r_{x} p + 42 \varphi _{p}^2 g_{py} r_{y} p^2 + 12 \varphi _{p}^2 g_{py} r^2\nonumber \\&\quad + 6 \varphi _{p}^2 g_{ppp} r^3 + 18 \varphi _{p}^2 g_{ppx} r^2 + 18 \varphi _{p}^2 g_{ppy} p r^2 - 54 \varphi _{p}^2 g_{pp} r_{p} r^2 - 6 \varphi _{p}^2 g_{pp} r_{x} r\nonumber \\&\quad - 6 \varphi _{p}^2 g_{pp} r_{y} p r - 24 \varphi _{p}^2 g_{p} r_{pp} r^2 + 30 \varphi _{p}^2 g_{p} r_{p}^2 r - 78 \varphi _{p}^2 g_{p} r_{p} r_{x} - 78 \varphi _{p}^2 g_{p} r_{p} r_{y} p\nonumber \\&\quad + 12 \varphi _{p}^2 g_{p} r_{xy} p + 6 \varphi _{p}^2 g_{p} r_{xx} + 6 \varphi _{p}^2 g_{p} r_{yy} p^2 - 2 \varphi _{p}^2 g_{p} r_{y} r - 18 \varphi _{p}^2 g_{xyy} p^2\nonumber \\&\quad + 96 \varphi _{p}^2 g_{xy} r_{p} p - 24 \varphi _{p}^2 g_{xy} r - 6 \varphi _{p}^2 g_{xxx} - 18 \varphi _{p}^2 g_{xxy} p + 48 \varphi _{p}^2 g_{xx} r_{p} + 24 \varphi _{p}^2 g_{x} r_{px}\nonumber \\&\quad + 24 \varphi _{p}^2 g_{x} r_{py} p + 30 \varphi _{p}^2 g_{x} r_{pp} r - 96 \varphi _{p}^2 g_{x} r_{p}^2 + 2 \varphi _{p}^2 g_{x} r_{y} - 6 \varphi _{p}^2 g_{yyy} p^3\nonumber \\&\quad + 48 \varphi _{p}^2 g_{yy} r_{p} p^2 - 24 \varphi _{p}^2 g_{yy} p r + 24 \varphi _{p}^2 g_{y} r_{px} p + 24 \varphi _{p}^2 g_{y} r_{py} p^2 + 30 \varphi _{p}^2 g_{y} r_{pp} p r\nonumber \\&\quad - 96 \varphi _{p}^2 g_{y} r_{p}^2 p + 36 \varphi _{p}^2 g_{y} r_{p} r + 24 \varphi _{p}^2 g_{y} r_{x} + 26 \varphi _{p}^2 g_{y} r_{y} p - 34 \varphi _{p} \varphi _{y} g_{px} r\nonumber \\&\quad - 34 \varphi _{p} \varphi _{y} g_{py} p r + 32 \varphi _{p} \varphi _{y} g_{pp} r^2 + 28 \varphi _{p} \varphi _{y} g_{p} r_{x} + 28 \varphi _{p} \varphi _{y} g_{p} r_{y} p - 44 \varphi _{p} \varphi _{y} g_{xy} p\nonumber \\&\quad - 22 \varphi _{p} \varphi _{y} g_{xx} + 86 \varphi _{p} \varphi _{y} g_{x} r_{p} - 22 \varphi _{p} \varphi _{y} g_{yy} p^2 + 86 \varphi _{p} \varphi _{y} g_{y} r_{p} p - 26 \varphi _{p} \varphi _{y} g_{y} r\nonumber \\&\quad - 5 \varphi _{y}^2 g_{p} r - 25 \varphi _{y}^2 g_{x} - 25 \varphi _{y}^2 g_{y} p)/((g_{x} + g_{y} p - g_{p} r) \varphi _{p}^2), \end{aligned}$$

(8.9)

$$\begin{aligned} C_1&= 15 (2 ((3 ((2 ((3 g_{xxy} + g_{yyy} p^2) p + g_{xxx} + 3 g_{xyy} p^2 - g_{ppy} p r^2 - g_{ppx} r^2) \nonumber \\&\quad + 3 (r_{x} + r_{y} p + r_{p} r) g_{pp} r - 2 (2 (r_{x} + r_{y} p) - 3 r_{p} r) g_{px}) r - (5 (r_{x} + r_{y} p)\nonumber \\&\quad + 11 r_{p} r) g_{xx} - (5 r_{y} p^2 - r^2 + 5 r_{x} p + 11 r_{p} p r) g_{yy} p - (10 r_{y} p^2 - r^2 + 10 r_{x} \nonumber \\&\quad \times p + 22 r_{p} p r) g_{xy}) - (12 r_{y} p^2 + 5 r^2 + 12 r_{x} p- 18 r_{p} p r) g_{py} r - (5 r_{yy} p^2 - 3 r_{y} r\nonumber \\&\quad + 5 r_{xx} + 10 r_{xy} p - 31 r_{p}^2 r + 8 r_{pp} r^2 + 14 r_{py} p r + 14 r_{px} r - 65 (r_{x} + r_{y} p) r_{p}) g_{x}\nonumber \\&\quad - ((5 r_{yy} p^2 + 6 r_{y} r) p + 9 r_{x} r + 5 r_{xx} p + 10 r_{xy} p^2 - 31 r_{p}^2 p r + 8 r_{pp} p r^2\nonumber \\&\quad + 14 r_{py} p^2 r + 14 r_{px} p r - (65 r_{y} p^2 + 3 r^2 + 65 r_{x} p)r_{p}) g_{y} + (15 (r_{x} + 2 r_{y} p) r_{x}\nonumber \\&\quad - (r_{yy} r - 15 r_{y}^2) p^2 - r_{xx} r - 2 r_{xy} p r - 22 r_{p}^2 r^2 + 8 r_{pp} r^3 + 8 r_{py} p r^2\nonumber \\&\quad + 8 r_{px} r^2 - 17 (r_{x} + r_{y} p) r_{p} r) g_{p}) \varphi _{p}^2 + ((22 (g_{xx} + g_{yy} p^2 + 2 g_{xy} p) - 9 g_{pp} r^2 \nonumber \\&\quad - 5 g_{py} p r - 5 g_{px} r - (3 (r_{x} + r_{y} p) - 32 r_{p} r) g_{p}) r - (25 (r_{x} + r_{y} p) + 61 r_{p} r) g_{x}\nonumber \\&\quad - (25 r_{y} p^2 - 2 r^2 + 25 r_{x} p + 61 r_{p} p r) g_{y}\varphi _{p} \varphi _{y}) + (2 (4 (3 (g_{x} + g_{y} p) \nonumber \\&\quad - 2 g_{p} r) \varphi _{py} \varphi _{p} - (g_{x} + g_{y} p - g_{p} r) (4 \varphi _{ppp} \varphi _{p} - 15 \varphi _{pp}^2) r) r + 5 (5 (g_{x} + g_{y} p)\nonumber \\&\quad - 2 g_{p} r) \varphi _{y}^2) r - (3 (3 ((3 (g_{xx} + g_{yy} p^2 + 2 g_{xy} p) - g_{pp} r^2 - 2 g_{py} p r - 2 g_{px} r\nonumber \\&\quad + (r_{x} + r_{y} p + 4 r_{p} r) g_{p}) r - (4 (r_{x} + r_{y} p) + 5 r_{p} r) g_{x}) - (12 r_{y} p^2 + r^2 + 12 r_{x} p\nonumber \\&\quad + 15 r_{p} p r) g_{y}) \varphi _{p} + 20 (3 (g_{x} + g_{y} p)- 2 g_{p} r) \varphi _{y} r) \varphi _{pp} r))/((g_{x} + g_{y} p - g_{p} r)\varphi _{p}^2), \end{aligned}$$

(8.10)

$$\begin{aligned} C_0&= - (12 \varphi _{py} \varphi _{p} g_{p} r^4 - 16 \varphi _{py} \varphi _{p} g_{x} r^3 - 16 \varphi _{py} \varphi _{p} g_{y} p r^3 - 4 \varphi _{ppp}\varphi _{p} g_{p} r^5 + 4 \varphi _{ppp} \varphi _{p} g_{x} r^4 \nonumber \\&\quad + 4 \varphi _{ppp} \varphi _{p} g_{y} p r^4 + 15 \varphi _{pp}^2 g_{p} r^5 - 15 \varphi _{pp}^2 g_{x} r^4 - 15 \varphi _{pp}^2 g_{y} p r^4 - 18 \varphi _{pp} \varphi _{p} g_{px} r^4 \nonumber \\&\quad - 18 \varphi _{pp} \varphi _{p} g_{py} p r^4 + 18 \varphi _{pp} \varphi _{p} g_{p} r_{p} r^4 + 18 \varphi _{pp} \varphi _{p} g_{p} r_{x} r^3 + 18 \varphi _{pp} \varphi _{p} g_{p} r_{y} p r^3\nonumber \\&\quad + 36 \varphi _{pp} \varphi _{p} g_{xy} p r^3 + 18 \varphi _{pp} \varphi _{p} g_{xx} r^3 - 18 \varphi _{pp} \varphi _{p} g_{x} r_{p} r^3 - 36 \varphi _{pp}\varphi _{p} g_{x} r_{x} r^2\nonumber \\&\quad - 36 \varphi _{pp} \varphi _{p} g_{x} r_{y} p r^2 + 18 \varphi _{pp} \varphi _{p} g_{yy} p^2 r^3 - 18 \varphi _{pp} \varphi _{p} g_{y} r_{p} p r^3 - 36 \varphi _{pp} \varphi _{p} g_{y} r_{x} p r^2\nonumber \\&\quad - 36 \varphi _{pp} \varphi _{p} g_{y} r_{y} p^2 r^2 - 6 \varphi _{pp} \varphi _{p} g_{y} r^4 - 30 \varphi _{pp} \varphi _{y} g_{p} r^4 + 40 \varphi _{pp} \varphi _{y} g_{x} r^3\nonumber \\&\quad + 40 \varphi _{pp} \varphi _{y} g_{y}p r^3 + 12 \varphi _{p}^2 g_{pxy} p r^3 + 6 \varphi _{p}^2 g_{pxx} r^3 - 18 \varphi _{p}^2 g_{px} r_{p} r^3 - 18 \varphi _{p}^2 g_{px} r_{x} r^2\nonumber \\&\quad - 18 \varphi _{p}^2 g_{px} r_{y} p r^2 + 6 \varphi _{p}^2 g_{pyy} p^2 r^3 - 18 \varphi _{p}^2 g_{py} r_{p} p r^3 - 18 \varphi _{p}^2 g_{py}r_{x}p r^2 \nonumber \\&\quad - 18 \varphi _{p}^2 g_{py} r_{y} p^2 r^2 - 4 \varphi _{p}^2 g_{p} r_{px} r^3 - 4 \varphi _{p}^2 g_{p} r_{py} p r^3 - 4 \varphi _{p}^2 g_{p} r_{pp} r^4 + 11 \varphi _{p}^2 g_{p} r_{p}^2 r^3\nonumber \\&\quad + 22 \varphi _{p}^2 g_{p} r_{p} r_{x} r^2 + 22 \varphi _{p}^2 g_{p} r_{p} r_{y} p r^2 - 8 \varphi _{p}^2 g_{p} r_{xy} p r^2 - 4 \varphi _{p}^2 g_{p} r_{xx} r^2 + 15 \varphi _{p}^2 g_{p} r_{x}^2 r\nonumber \\&\quad + 30 \varphi _{p}^2 g_{p} r_{x} r_{y} p r - 4 \varphi _{p}^2 g_{p} r_{yy} p^2 r^2 + 15\varphi _{p}^2 g_{p} r_{y}^2 p^2 r + 4 \varphi _{p}^2 g_{p} r_{y} r^3 - 18 \varphi _{p}^2 g_{xyy} p^2 r^2\nonumber \\&\quad + 36 \varphi _{p}^2 g_{xy} r_{p} p r^2 + 60 \varphi _{p}^2 g_{xy} r_{x} p r + 60 \varphi _{p}^2 g_{xy} r_{y} p^2 r + 4 \varphi _{p}^2 g_{xy} r^3 - 6 \varphi _{p}^2 g_{xxx} r^2\nonumber \\&\quad - 18 \varphi _{p}^2 g_{xxy} p r^2 + 18 \varphi _{p}^2 g_{xx} r_{p} r^2 + 30 \varphi _{p}^2 g_{xx} r_{x} r + 30 \varphi _{p}^2 g_{xx} r_{y} p r + 4 \varphi _{p}^2 g_{x} r_{px} r^2\nonumber \\&\quad + 4 \varphi _{p}^2 g_{x} r_{py} p r^2 + 4 \varphi _{p}^2 g_{x} r_{pp} r^3 - 11 \varphi _{p}^2 g_{x} r_{p}^2 r^2 - 40 \varphi _{p}^2 g_{x} r_{p} r_{x} r - 40 \varphi _{p}^2 g_{x} r_{p} r_{y} p r\nonumber \\&\quad + 20 \varphi _{p}^2 g_{x} r_{xy} p r + 10 \varphi _{p}^2 g_{x} r_{xx} r - 45 \varphi _{p}^2 g_{x} r_{x}^2 - 90 \varphi _{p}^2 g_{x} r_{x} r_{y} p + 10 \varphi _{p}^2 g_{x} r_{yy} p^2 r \nonumber \\&\quad - 45 \varphi _{p}^2 g_{x} r_{y}^2 p^2 - 8 \varphi _{p}^2 g_{x} r_{y} r^2 - 6 \varphi _{p}^2 g_{yyy} p^3 r^2 + 18 \varphi _{p}^2 g_{yy} r_{p} p^2 r^2 + 30 \varphi _{p}^2 g_{yy} r_{x} p^2 r \nonumber \\&\quad + 30 \varphi _{p}^2 g_{yy} r_{y} p^3 r + 4 \varphi _{p}^2 g_{yy} p r^3 + 4 \varphi _{p}^2 g_{y} r_{px} p r^2 + 4 \varphi _{p}^2 g_{y} r_{py} p^2 r^2 + 4 \varphi _{p}^2 g_{y} r_{pp} p r^3\nonumber \\&\quad - 11 \varphi _{p}^2 g_{y} r_{p}^2 p r^2 - 40 \varphi _{p}^2 g_{y} r_{p} r_{x} p r - 40 \varphi _{p}^2 g_{y} r_{p} r_{y} p^2 r - 6 \varphi _{p}^2 g_{y} r_{p} r^3 + 20 \varphi _{p}^2 g_{y} r_{xy} p^2 r\nonumber \\&\quad + 10 \varphi _{p}^2 g_{y} r_{xx} p r- 45 \varphi _{p}^2 g_{y} r_{x}^2 p - 90 \varphi _{p}^2 g_{y} r_{x} r_{y} p^2 - 6 \varphi _{p}^2 g_{y} r_{x} r^2 + 10 \varphi _{p}^2 g_{y} r_{yy} p^3 r\nonumber \\&\quad - 45 \varphi _{p}^2 g_{y} r_{y}^2 p^3 - 14 \varphi _{p}^2 g_{y} r_{y} p r^2 + 18 \varphi _{p} \varphi _{y} g_{px} r^3 + 18 \varphi _{p} \varphi _{y} g_{py} p r^3 - 26 \varphi _{p} \varphi _{y} g_{p} r_{p} r^3\nonumber \\&\quad - 22 \varphi _{p} \varphi _{y} g_{p} r_{x} r^2 - 22 \varphi _{p} \varphi _{y} g_{p} r_{y} p r^2 - 44 \varphi _{p} \varphi _{y} g_{xy} p r^2 - 22 \varphi _{p} \varphi _{y} g_{xx} r^2 + 36 \varphi _{p} \varphi _{y} g_{x} r_{p} r^2\nonumber \\&\quad + 50 \varphi _{p} \varphi _{y} g_{x} r_{x} r + 50 \varphi _{p} \varphi _{y} g_{x} r_{y} p r - 22 \varphi _{p} \varphi _{y} g_{yy} p^2 r^2 + 36 \varphi _{p} \varphi _{y} g_{y} r_{p} p r^2 \nonumber \\&\quad + 50 \varphi _{p} \varphi _{y} g_{y} r_{x} p r + 50 \varphi _{p} \varphi _{y} g_{y} r_{y} p^2 r + 6 \varphi _{p} \varphi _{y} g_{y} r^3 + 15 \varphi _{y}^2 g_{p} r^3 - 25 \varphi _{y}^2 g_{x} r^2 \nonumber \\&\quad - 25 \varphi _{y}^2 g_{y} p r^2)/((g_{x} + g_{y} p - g_{p} r) \varphi _{p}^2), \end{aligned}$$

(8.11)

$$\begin{aligned} D_7&=((\varphi _{pppp} \varphi _{p} g_{p} - 10 \varphi _{ppp} \varphi _{pp} g_{p} + 4 \varphi _{ppp} \varphi _{p} g_{pp}) \varphi _{p} + 15 \varphi _{pp}^3 g_{p} - 15 \varphi _{pp}^2 \varphi _{p} g_{pp}\nonumber \\&\quad + 6 \varphi _{pp} \varphi _{p}^2 g_{ppp} - \varphi _{p}^3 g_{pppp})/((g_{x} + g_{y} p - g_{p} r) \varphi _{p}^3), \end{aligned}$$

(8.12)

$$\begin{aligned} D_6&= - (6 ((3 (2 g_{pp} r_{p} + g_{p} r_{pp} - g_{ppy} p- g_{ppx}) - 4 g_{ppp} r - 3 g_{py}) \varphi _{p} - 5 \varphi _{y} g_{pp}) \varphi _{pp} \varphi _{p}\nonumber \\&\quad - (15 ((2 g_{p} r_{p} - g_{y} - 5 g_{pp} r - 2 g_{py} p - 2 g_{px}) \varphi _{p} - 3 \varphi _{y} g_{p}) \varphi _{pp}^2 - (((3 g_{pppp} r\nonumber \\&\quad + 4 g_{pppx} + 4 g_{pppy} p - 12 g_{ppp} r_{p} + 6 g_{ppy} - 12 g_{pp} r_{pp} - 4 g_{p} r_{ppp}) \varphi _{p} + 6 \varphi _{y} g_{ppp})\nonumber \\&\quad \times \varphi _{p}^2 - 15 (g_{x} + g_{y} p + 6 g_{p} r) \varphi _{pp}^3)) + 6 \varphi _{ppy} \varphi _{p}^2 g_{p} + 2 (2 (2 g_{p} r_{p} - g_{y - 5 g_{pp} r}\nonumber \\&\quad - 2 g_{py} p - 2 g_{px}) \varphi _{p} - 5 \varphi _{y} g_{p} + 5 (g_{x} + g_{y} p + 6 g_{p} r) \varphi _{pp}) \varphi _{ppp} \varphi _{p} - (g_{x} + g_{y} p\nonumber \\&\quad + 6 g_{p} r) \varphi _{pppp} \varphi _{p}^2 - 6 (5 \varphi _{pp} g_{p} - 2 \varphi _{p} g_{pp}) \varphi _{py} \varphi _{p})/((g_{x} + g_{y} p - g_{p} r) \varphi _{p}^3), \end{aligned}$$

(8.13)

$$\begin{aligned} D_5&=((4 (r_{ppp} p + 3 r_{pp}) g_{y} - 3 g_{yy} + 4 g_{x} r_{ppp} + 2 (3 (r_{ppy} p - 7 r_{pp} r_{p} + r_{ppx})+ 7 r_{ppp} r) g_{p}\nonumber \\&\quad + 12 (3 r_{pp} r - 4 r_{p}^2 + r_{py} p + r_{px}) g_{pp} + 18 (2 r_{p} p - r) g_{ppy} - 6 g_{ppyy} p^2 + 36 g_{ppx} r_{p}\nonumber \\&\quad - 6 g_{ppxx} - 12 g_{ppxy} p + 6 (r_{x} + r_{y} p + 5 r_{p} r) g_{ppp} - 12 g_{pppy} p r - 12 g_{pppx} r - 3 g_{pppp} r^2\nonumber \\&\quad + 12 (2 r_{pp} p + 3 r_{p}) g_{py} - 12 g_{pyy} p + 24 g_{px} r_{pp} - 12 g_{pxy}) \varphi _{p}^2 + 3 ((\varphi _{yy} g_{p} - 6 \varphi _{y} g_{py}\nonumber \\&\quad - 6 \varphi _{y} g_{ppp} r - 6 \varphi _{y} g_{ppx} - 6 \varphi _{y} g_{ppy} p + 16 \varphi _{y} g_{pp} r_{p} + 8 \varphi _{y} g_{p} r_{pp}) \varphi _{p} - 5 \varphi _{y}^2 g_{pp})) \varphi _{p}\nonumber \\&\quad - 3 (6 ((r_{pp} p + 2 r_{p}) g_{y} - g_{yy} p + g_{x} r_{pp} - g_{xy} + (4 r_{pp} r - 3 r_{p}^2 + r_{py} p + r_{px})g_{p}\nonumber \\&\quad + (r_{x} + r_{y} p + 7 r_{p} r) g_{pp} - 4 g_{ppy} p r - 4 g_{ppx} r - 2 g_{ppp} r^2 + 4 (r_{p} p - r) g_{py} -g_{pyy} p^2\nonumber \\&\quad + 4 g_{px} r_{p} - g_{pxx} - 2 g_{pxy} p) \varphi _{p}^2 + 5 (2 (3 g_{p} r_{p} - g_{y} - 4 g_{pp} r - 2 g_{py} p - 2 g_{px}) \varphi _{p}\nonumber \\&\quad - 3 \varphi _{y} g_{p}) \varphi _{y} - 5 (((2 r_{p} p - 5 r) g_{y} - g_{yy} p^2 + 2 g_{x} r_{p} - g_{xx} - 2 g_{xy} p + (r_{x} + r_{y} p\nonumber \\&\quad + 9 r_{p} r) g_{p} - 10 g_{pp} r^2 - 10 g_{py} p r - 10 g_{px} r) \varphi _{p} - 3 (g_{x} + g_{y} p + 5 g_{p} r) \varphi _{y} + 3 (2 (g_{x}\nonumber \\&\quad + g_{y} p) + 5 g_{p} r) \varphi _{pp} r) \varphi _{pp}) \varphi _{pp} - 6 (g_{x} + g_{y} p + 5 g_{p} r) \varphi _{ppy} \varphi _{p}^2 - 2 (2 ((2 r_{p} p - 5 r) g_{y}\nonumber \\&\quad - g_{yy} p^2 + 2 g_{x} r_{p} - g_{xx} - 2 g_{xy} p + (r_{x} + r_{y} p + 9 r_{p} r) g_{p} - 10 g_{pp} r^2 - 10 g_{py} p r\nonumber \\&\quad - 10 g_{px} r) \varphi _{p} - 5 (g_{x} + g_{y} p + 5 g_{p} r) \varphi _{y} + 15 (2 (g_{x} + g_{y} p) + 5 g_{p} r) \varphi _{pp} r) \varphi _{ppp} \varphi _{p}\nonumber \\&\quad + 3 (2 (g_{x} + g_{y} p) + 5 g_{p} r) \varphi _{pppp} \varphi _{p}^2 r + 6 (2 (3 g_{p} r_{p} - g_{y} - 4 g_{pp} r - 2 g_{py} p - 2 g_{px}) \varphi _{p}\nonumber \\&\quad - 5 \varphi _{y} g_{p} + 5 (g_{x} + g_{y} p + 5 g_{p} r) \varphi _{pp}) \varphi _{py} \varphi _{p}/((g_{x} + g_{y} p - g_{p} r) \varphi _{p}^3), \end{aligned}$$

(8.14)

$$\begin{aligned} D_4&=(2 ((32 r_{p}^2 - r_{y}) r_{p} + r_{xy} + r_{yy} p - (11 (r_{x} + r_{y} p) + 49 r_{p} r) r_{pp} + 8 r_{ppy} p r +8 r_{ppx} r\nonumber \\&\quad + 10 r_{ppp} r^2 - 3 (8 r_{p} p + r) r_{py} + 2 r_{pyy} p^2 - 24 r_{px} r_{p} + 2 r_{pxx} + 4 r_{pxy} p)g_{p} + 24 g_{xy} r_{pp} p\nonumber \\&\quad + 36 g_{xy} r_{p} - 6 g_{xxy} + 12 g_{xx} r_{pp} + 14 g_{x} r_{ppp} r + 6 g_{x} r_{ppx} + 6 g_{x} r_{ppy} p - 42 g_{x} r_{pp} r_{p}\nonumber \\&\quad - 6 g_{yyy} p^2 + 12 g_{yy} r_{pp} p^2 + 36 g_{yy} r_{p} p - 9 g_{yy} r + 12 g_{y} r_{px} + 12 g_{y} r_{py} p + 14 g_{y} r_{ppp} p r\nonumber \\&\quad + 6 g_{y} r_{ppx} p + 6 g_{y} r_{ppy} p^2 - 42 g_{y} r_{pp} r_{p} p + 36 g_{y} r_{pp} r - 48 g_{y} r_{p}^2 - 12 g_{xyy} p\nonumber \\&\quad + 4 (r_{yy} p^2 - r_{y} r + r_{xx} + 2 r_{xy} p - (13 (r_{x} + r_{y} p) + 23 r_{p} r) r_{p} + 10 r_{pp} r^2 + 7 r_{py} p r\nonumber \\&\quad + 7 r_{px} r) g_{pp} + 18 (r_{y} p^2- r^2 + r_{x} p + 5 r_{p} p r) g_{ppy} - 18 g_{ppyy} p^2 r + 18 (r_{x} + r_{y} p\nonumber \\&\quad + 5 r_{p} r) g_{ppx} - 18 g_{ppxx} r - 36 g_{ppxy} p r + 12 (r_{x} + r_{y} p + 2 r_{p} r) g_{ppp} r - 12 g_{pppy} p r^2\nonumber \\&\quad - 12 g_{pppx} r^2 - g_{pppp} r^3 - 6 ((16 r_{p} p - 15 r) r_{p} - 3 (r_{x} + r_{y} p) - 12 r_{pp} p r - 4 r_{py} p^2\nonumber \\&\quad - 4 r_{px} p) g_{py} + 36 (r_{p} p - r) g_{pyy} p - 4 g_{pyyy} p^3 + 24 (3 r_{pp} r - 4 r_{p}^2 + r_{py} p + r_{px}) g_{px}\nonumber \\&\quad + 36 g_{pxx} r_{p} - 12 g_{pxxy} p - 4 g_{pxxx} + 36 (2 r_{p} p - r) g_{pxy} - 12 g_{pxyy} p^2) \varphi _{p}^3 \nonumber \\&\quad +(2 ((11 (r_{x} + r_{y} p) + 61 r_{p} r) g_{pp} + 11 g_{p} r_{px} + 11 g_{p} r_{py} p + 37 g_{p} r_{pp} r - 47 g_{p} r_{p}^2\nonumber \\&\quad - g_{p} r_{y} - 9 g_{xy} + 12 g_{x} r_{pp} - 9 g_{yy} p + 12 g_{y} r_{pp} p + 24 g_{y} r_{p} - 27 g_{ppy} p r - 27 g_{ppx} r\nonumber \\&\quad - 9 g_{ppp} r^2 + 3 (16 r_{p} p - 9 r) g_{py} - 9 g_{pyy} p^2 + 48 g_{px} r_{p} - 9 g_{pxx} - 18 g_{pxy} p) \varphi _{y}\nonumber \\&\quad + 3 (g_{x} + g_{y} p + 4 g_{p} r) \varphi _{yy}) \varphi _{p}^2 + 15 ((4 g_{p} r_{p} - g_{y} - 3 g_{pp} r - 2 g_{py} p - 2 g_{px}) \varphi _{p}\nonumber \\&\quad - \varphi _{y} g_{p}) \varphi _{y}^2 + (6 (3 (((3 r_{p} p - 7 r) r_{p} - (r_{x} + r_{y} p) - 4 r_{pp} p r - r_{py} p^2 - r_{px} p) g_{y}\nonumber \\&\quad - 2 (r_{p} p - 2 r) g_{yy} p) + g_{yyy} p^3 - 3 (4 r_{pp} r - 3 r_{p}^2 + r_{py} p + r_{px}) g_{x} - 6 g_{xx} r_{p}\nonumber \\&\quad + 3 g_{xxy} p + g_{xxx} - 12 (r_{p} p - r) g_{xy} + 3 g_{xyy} p^2 - (r_{yy} p^2 - r_{y} r + r_{xx} + 2 r_{xy} p \nonumber \\&\quad - 2 (5 (r_{x} + r_{y} p) + 13 r_{p} r) r_{p} + 19 r_{pp} r^2 + 10 r_{py} p r + 10 r_{px} r) g_{p} - 9 (r_{x} + r_{y} p\nonumber \\&\quad + 3 r_{p} r) g_{pp} r + 18 g_{ppy} p r^2 + 18 g_{ppx} r^2 + 4 g_{ppp} r^3 - 6 (r_{y} p^2 - 3 r^2 + r_{x} p + 7 r_{p} p r) g_{py}\nonumber \\&\quad + 12 g_{pyy} p^2 r - 6 (r_{x} + r_{y} p + 7 r_{p} r) g_{px} + 12 g_{pxx} r + 24 g_{pxy} p r) \varphi _{p}^2- 5 (2 (3 ((3 r_{p} p \nonumber \\&\quad - 4 r) g_{y} - g_{yy} p^2 + 3 g_{x} r_{p} - g_{xx} - 2 g_{xy} p) + 4 (r_{x} + r_{y} p + 8 r_{p} r) g_{p} - 18 g_{pp} r^2\nonumber \\&\quad - 24 g_{py} p r - 24 g_{px} r) \varphi _{p} - 9 (g_{x} + g_{y} p + 4 g_{p} r) \varphi _{y}) \varphi _{y} + 15 (((r_{y} p^2 - 10 r^2 + r_{x} p\nonumber \\&\quad + 9 r_{p} p r) g_{y} - 5 g_{yy} p^2 r + (r_{x} + r_{y} p + 9 r_{p} r) g_{x} - 5 g_{xx} r - 10 g_{xy} p r + 4 (r_{x} + r_{y} p\nonumber \\&\quad + 4 r_{p} r) g_{p} r - 10 g_{pp} r^3 - 20 g_{py} p r^2 - 20 g_{px} r^2) \varphi _{p} - 15 (g_{x} + g_{y} p + 2 g_{p} r) \varphi _{y} r\nonumber \\&\quad + 5 (3 (g_{x} + g_{y} p) + 4 g_{p} r) \varphi _{pp} r^2) \varphi _{pp}) \varphi _{pp} - 30 (g_{x} + g_{y} p + 2 g_{p} r) \varphi _{ppy} \varphi _{p}^2 r\nonumber \\&\quad - 2 (2 ((r_{y} p^2 - 10 r^2 + r_{x} p + 9 r_{p} p r) g_{y} - 5 g_{yy} p^2 r + (r_{x} + r_{y} p + 9 r_{p} r) g_{x} - 5 g_{xx} r \nonumber \\&\quad - 10 g_{xy} p r + 4 (r_{x} + r_{y} p + 4 r_{p} r) g_{p} r - 10 g_{pp} r^3 - 20 g_{py} p r^2 - 20 g_{px} r^2) \varphi _{p} - 25 (g_{x} \nonumber \\&\quad + g_{y} p + 2 g_{p} r) \varphi _{y} r + 25 (3 (g_{x} + g_{y} p) + 4 g_{p} r) \varphi _{pp} r^2) \varphi _{ppp} \varphi _{p} + 5 (3 (g_{x} + g_{y} p)\nonumber \\&\quad + 4 g_{p} r) \varphi _{pppp} \varphi _{p}^2 r^2 + 2 (2 (3 ((3 r_{p} p - 4 r) g_{y} - g_{yy} p^2 + 3 g_{x} r_{p} - g_{xx} - 2 g_{xy} p)\nonumber \\&\quad + 4 (r_{x} + r_{y} p + 8 r_{p} r) g_{p} - 18 g_{pp} r^2 - 24 g_{py} p r - 24 g_{px} r) \varphi _{p} - 15 (g_{x} + g_{y} p \nonumber \\&\quad + 4 g_{p} r) \varphi _{y} + 75 (g_{x} + g_{y} p + 2 g_{p} r) \varphi _{pp} r) \varphi _{py} \varphi _{p}/((g_{x} + g_{y} p - g_{p} r) \varphi _{p}^3), \end{aligned}$$

(8.15)

$$\begin{aligned} D_3&=((2 (3 r_{yy} p^2 - 2 r_{y} r + 2 r_{xx} + 5 r_{xy} p +(2 (16 r_{p} p - 23 r) r_{p} - (26 r_{x} + 27 r_{y} p)) r_{p}\\&\quad - (11 r_{y} p^2 - 20 r^2 + 11 r_{x} p + 49 r_{p} p r) r_{pp} + 8 r_{ppy} p^2 r + 8 r_{ppx} p r + 10 r_{ppp} p r^2\\&\quad - (24 r_{p} p - 11 r) r_{py} p + 2 r_{pyy} p^3 - 2 (12 r_{p} p - 7 r) r_{px} + 2 r_{pxx} p + 4 r_{pxy} p^2) g_{y}\\&\quad + 3 (3 (2 r_{y} p^2 - r^2 + 2 r_{x} p) - 2 (8 r_{p} p - 15 r) r_{p} p + 12 r_{pp} p^2 r + 4 r_{py} p^3\\&\quad + 4 r_{px} p^2) g_{yy} + 6 (2 r_{p} p - 3 r) g_{yyy} p^2 - g_{yyyy} p^4 + 2 ((32 r_{p}^2 - r_{y}) r_{p} + r_{xy} + r_{yy} p\\&\quad - (11 (r_{x} + r_{y} p) + 49 r_{p} r) r_{pp} + 8 r_{ppy} p r + 8 r_{ppx} r + 10 r_{ppp} r^2 - 3 (8 r_{p} p + r) r_{py}\\&\quad + 2 r_{pyy} p^2 - 24 r_{px} r_{p} + 2 r_{pxx} + 4 r_{pxy} p) g_{x} + 12 (3 r_{pp} r - 4 r_{p}^2 + r_{py} p + r_{px}) g_{xx}\\&\quad + 18 (2 r_{p} p - r) g_{xxy} - 6 g_{xxyy} p^2 + 12 g_{xxx} r_{p} - 4 g_{xxxy} p - g_{xxxx} - 6 ((16 r_{p} p\\&\quad - 15 r) r_{p} - 3 (r_{x} + r_{y} p) - 12 r_{pp} p r - 4 r_{py} p^2 - 4 r_{px} p) g_{xy} + 36 (r_{p} p - r) g_{xyy} p\\&\quad - 4 g_{xyyy} p^3 + ((3 (r_{yy} r - r_{y}^2) + r_{yyy} p^2) p - 3 r_{x} r_{y} + 3 r_{xxy} p + r_{xxx} + 3 r_{xy} r \\&\quad + 3 r_{xyy} p^2 - (17 r_{yy} p^2 - 14 r_{y} r + 17 r_{xx} + 34 r_{xy} p - (113 (r_{x} + r_{y} p)\\&\quad + 79 r_{p} r) r_{p}) r_{p} - (39 (r_{x} + r_{y} p) + 86 r_{p} r) r_{pp} r + 15 r_{ppy} p r^2 + 15 r_{ppx} r^2 + 15 r_{ppp} r^3\\&\quad - (83 r_{p} p r + 27 r_{x} p + 27 r_{y} p^2 + 15 r^2) r_{py} + 9 r_{pyy} p^2 r - (27 (r_{x} + r_{y} p) + 83 r_{p} r) r_{px}\\&\quad + 9 r_{pxx} r + 18 r_{pxy} p r) g_{p} - ((15 r_{y} p^2 + 8 r^2) r_{y} - 8 r_{yy} p^2 r + 15 (r_{x} + 2 r_{y} p) r_{x}\\&\quad - 8 r_{xx} r - 16 r_{xy} p r + (74 (r_{x} + r_{y} p) + 55 r_{p} r) r_{p} r - 20 r_{pp} r^3 - 20 r_{py} p r^2\\&\quad - 20 r_{px} r^2) g_{pp} + 6 (6 r_{y} p^2 - r^2 + 6 r_{x} p + 12 r_{p} p r) g_{ppy} r - 18 g_{ppyy} p^2 r^2\\&\quad + 36 (r_{x} + r_{y} p + 2 r_{p} r) g_{ppx} r - 18 g_{ppxx} r^2 - 36 g_{ppxy} p r^2 + 6 (r_{x} + r_{y} p + r_{p} r) g_{ppp} r^2\\&\quad - 4 g_{pppy} p r^3 - 4 g_{pppx} r^3+ 4 ((2 r_{yy} p^2 + 7 r_{y} r) p + 9 r_{x} r + 2 r_{xx} p + 4 r_{xy} p^2 - 2 (13 r_{y} p^2\\&\quad - 9 r^2 + 13 r_{x} p + 23 r_{p} p r) r_{p} + 20 r_{pp} p r^2 + 14 r_{py} p^2 r + 14 r_{px} p r) g_{py} + 18 (r_{y} p^2\\&\quad - 2 r^2 + r_{x} p + 5 r_{p} p r) g_{pyy} p - 12 g_{pyyy} p^3 r + 8 (r_{yy} p^2 - r_{y} r + r_{xx} + 2 r_{xy} p\\&\quad - (13 (r_{x} + r_{y} p) + 23 r_{p} r) r_{p} + 10 r_{pp} r^2 + 7 r_{py} p r + 7 r_{px} r) g_{px} + 18 (r_{x} + r_{y} p\\&\quad + 5 r_{p} r) g_{pxx} - 36 g_{pxxy} p r - 12 g_{pxxx} r + 36 (r_{y} p^2 - r^2 + r_{x} p + 5 r_{p} p r) g_{pxy}\\&\quad - 36 g_{pxyy} p^2 r) \varphi _{p}^3 - (((2 (((47 r_{p} p - 61 r) r_{p} - (11 r_{x} + 10 r_{y} p) - 37 r_{pp} p r - 11 r_{py} p^2\\&\quad - 11 r_{px} p) g_{y} - 3 (8 r_{p} p - 9 r) g_{yy} p + 3 g_{yyy} p^3 + (47 r_{p}^2 + r_{y} - 37 r_{pp} r - 11 r_{py} p\\&\quad - 11 r_{px}) g_{x} - 24 g_{xx} r_{p} + 9 g_{xxy} p + 3 g_{xxx} - 3 (16 r_{p} p - 9 r) g_{xy} + 9 g_{xyy} p^2)\\ \end{aligned}$$

$$\begin{aligned}&\quad \quad \;\;- (7 r_{yy} p^2 - 13 r_{y} r + 7 r_{xx} + 14 r_{xy} p - 3 (31 (r_{x} + r_{y} p) + 63 r_{p} r) r_{p} + 85 r_{pp} r^2\nonumber \\&\quad \quad \;\;+ 52 r_{py} p r + 52 r_{px} r) g_{p} - 4 (11 (r_{x} + r_{y} p) + 25 r_{p} r) g_{pp} r + 54 g_{ppy} p r^2\nonumber \\&\quad \quad \;\;+ 54 g_{ppx} r^2 + 6 g_{ppp} r^3 - 2 (22 r_{y} p^2 - 27 r^2 + 22 r_{x} p + 122 r_{p} p r) g_{py} + 54 g_{pyy} p^2 r\nonumber \\&\quad \quad \;\;- 4 (11 (r_{x} + r_{y} p) + 61 r_{p} r) g_{px} + 54 g_{pxx} r + 108 g_{pxy} p r) \varphi _{y} - 6 (2 (g_{x} + g_{y} p)\nonumber \\&\quad \quad \;\;+ 3 g_{p} r) \varphi _{yy} r) \varphi _{p}^2 - 5 ((3 ((4 r_{p} p - 3 r) g_{y} - g_{yy} p^2 + 4 g_{x} r_{p} - g_{xx} - 2 g_{xy} p)\nonumber \\&\quad \quad \;\;+ (5 (r_{x} + r_{y} p) + 31 r_{p} r) g_{p} - 9 g_{pp} r^2 - 18 g_{py} p r - 18 g_{px} r) \varphi _{p} - 3 (g_{x} + g_{y} p \nonumber \\&\quad \quad \;\;+ 3 g_{p} r) \varphi _{y}) \varphi _{y}^2) - (2 (3 (((r_{yy} p^2 + 8 r_{y} r) p + 9 r_{x} r + r_{xx} p + 2 r_{xy} p^2 - (10 r_{y} p^2 - 27 r^2 \nonumber \\&\quad \quad \;\;+ 10 r_{x} p + 26 r_{p} p r) r_{p} + 19 r_{pp} p r^2 + 10 r_{py} p^2 r + 10 r_{px} p r) g_{y} + 3 (r_{y} p^2 - 6 r^2 + r_{x} p \nonumber \\&\quad \quad \;\;+ 7 r_{p} p r) g_{yy} p - 4 g_{yyy} p^3 r + (r_{yy} p^2 - r_{y} r + r_{xx} + 2 r_{xy} p - 2 (5 (r_{x} + r_{y} p) + 13 r_{p} r) r_{p} \nonumber \\&\quad \quad \;\;+ 19 r_{pp} r^2 + 10 r_{py} p r + 10 r_{px} r) g_{x} + 3 (r_{x} + r_{y} p + 7 r_{p} r) g_{xx} - 12 g_{xxy} p r - 4 g_{xxx} r \nonumber \\&\quad \quad \;\;+ 6 (r_{y} p^2 - 3 r^2 + r_{x} p + 7 r_{p} p r) g_{xy} - 12 g_{xyy} p^2 r - 3 ((r_{y} p^2 + r^2) r_{y} - r_{yy} p^2 r + (r_{x} \nonumber \\&\quad \quad \;\;+ 2 r_{y} p) r_{x} - r_{xx} r - 2 r_{xy} p r + (8 (r_{x} + r_{y} p) + 9 r_{p} r) r_{p} r - 5 r_{pp} r^3 - 4 r_{py} p r^2 - 4 r_{px} r^2) g_{p} \nonumber \\&\quad \quad \;\;+ 3 (3 (r_{x} + r_{y} p) + 5 r_{p} r) g_{pp} r^2 - 12 g_{ppy} p r^3 - 12 g_{ppx} r^3 - g_{ppp} r^4+ 6 (3 r_{y} p^2 - 2 r^2 \nonumber \\&\quad \quad \;\;+ 3 r_{x} p + 9 r_{p} p r) g_{py} r - 18 g_{pyy} p^2 r^2 + 18 (r_{x} + r_{y} p + 3 r_{p} r) g_{px} r - 18 g_{pxx} r^2 \nonumber \\&\quad \quad \;\;- 36 g_{pxy} p r^2) \varphi _{p}^2 + 5 (2 ((2 r_{y} p^2 - 9 r^2+ 2 r_{x} p + 16 r_{p} p r) g_{y} - 6 g_{yy} p^2 r + 2 (r_{x} + r_{y} p \nonumber \\&\quad \quad \;\;+ 8 r_{p} r) g_{x} - 6 g_{xx} r- 12 g_{xy} p r + 3 (2 (r_{x} + r_{y} p) + 7 r_{p} r) g_{p} r- 6 g_{pp} r^3 - 18 g_{py} p r^2 \nonumber \\&\quad \quad \;\;- 18 g_{px} r^2) \varphi _{p} - 9 (2 (g_{x} + g_{y} p) + 3 g_{p} r) \varphi _{y} r) \varphi _{y}) - 15 ((2 ((2 r_{y} p^2 - 5 r^2 + 2 r_{x} p \nonumber \\&\quad \quad \;\;+ 8 r_{p} p r) g_{y} - 5 g_{yy} p^2 r + 2 (r_{x} + r_{y} p + 4 r_{p} r) g_{x} - 5 g_{xx} r - 10 g_{xy} p r + (3 (r_{x} + r_{y} p) \nonumber \\&\quad \quad \;\;+ 7 r_{p} r) g_{p} r) - 5 g_{pp} r^3 - 20 g_{py} p r^2 - 20 g_{px} r^2) \varphi _{p} - 30 (g_{x} + g_{y} p + g_{p} r) \varphi _{y} r + 5 (4 (g_{x} \nonumber \\&\quad \quad \;\;+ g_{y} p) + 3 g_{p} r) \varphi _{pp} r^2) \varphi _{pp} r) \varphi _{pp} - 60 (g_{x} + g_{y} p + g_{p} r) \varphi _{ppy} \varphi _{p}^2 r^2 - 2 (2 ((2 ((2 r_{y} p^2 \nonumber \\&\quad \quad \;\;- 5 r^2 + 2 r_{x} p + 8 r_{p} p r) g_{y} - 5 g_{yy} p^2 r + 2 (r_{x} + r_{y} p + 4 r_{p} r) g_{x} - 5 g_{xx} r - 10 g_{xy} p r\nonumber \\&\quad \quad \;\;+ (3 (r_{x} + r_{y} p) + 7 r_{p} r) g_{p} r) - 5 g_{pp} r^3 - 20 g_{py} p r^2 - 20 g_{px} r^2) \varphi _{p} - 25 (g_{x} + g_{y} p\nonumber \\&\quad \quad \;\;+ g_{p} r) \varphi _{y} r) + 25 (4 (g_{x} + g_{y} p) + 3 g_{p} r) \varphi _{pp} r^2) \varphi _{ppp} \varphi _{p} r + 5 (4 (g_{x} + g_{y} p)\nonumber \\&\quad \quad \;\;+ 3 g_{p} r) \varphi _{pppp} \varphi _{p}^2 r^3 + 4 (2 ((2 r_{y} p^2 - 9 r^2 + 2 r_{x} p + 16 r_{p} p r) g_{y} - 6 g_{yy} p^2 r + 2 (r_{x}\nonumber \\&\quad \quad \;\;+ r_{y} p + 8 r_{p} r) g_{x} - 6 g_{xx} r - 12 g_{xy} p r + 3 (2 (r_{x} + r_{y} p) + 7 r_{p} r) g_{p} r - 6 g_{pp} r^3 \nonumber \\&\quad \quad \;\;- 18 g_{py} p r^2 - 18 g_{px} r^2) \varphi _{p} - 15 (2 (g_{x} + g_{y} p) + 3 g_{p} r) \varphi _{y} r + 75 (g_{x} + g_{y} p\nonumber \\&\quad \quad \;\;+ g_{p} r) \varphi _{pp} r^2) \varphi _{py} \varphi _{p})/((g_{x} + g_{y} p -g_{p} r) \varphi _{p}^3), \end{aligned}$$

(8.16)

$$\begin{aligned} D_2&= - (((2 (9 r_{y} p^2 + 4 r^2) r_{y} - 11 r_{yy} p^2 r -r_{yyy} p^4 + 3 (5 r_{x} + 11 r_{y} p) r_{x} - 8 r_{xx} r - 3 r_{xxy} p^2\\&\quad \;\,- r_{xxx} p - 19 r_{xy} p r - 3 r_{xyy} p^3 + ((17 r_{yy} p^2 + 60 r_{y} r) p + 74 r_{x} r + 17 r_{xx} p + 34 r_{xy} p^2\\&\quad \;\,- (113 r_{y} p^2 - 55 r^2 + 113 r_{x} p + 79 r_{p} p r) r_{p}) r_{p} + (39 r_{y} p^2 - 20 r^2 + 39 r_{x} p\\&\quad \;\,+ 86 r_{p} p r) r_{pp} r - 15 r_{ppy} p^2 r^2 - 15 r_{ppx} p r^2 - 15 r_{ppp} p r^3 + (27 r_{y} p^2 - 5 r^2\\&\quad \;\,+ 27 r_{x} p + 83 r_{p} p r) r_{py} p - 9 r_{pyy} p^3 r + (27 r_{y} p^2 - 20 r^2 + 27 r_{x} p + 83 r_{p} p r) r_{px}\\&\quad \;\,- 9 r_{pxx} p r - 18 r_{pxy} p^2 r) g_{y} - ((32 r_{y} p^2 - 3 r^2) r + 4 r_{yy} p^4 + 36 r_{x} p r + 4 r_{xx} p^2\\&\quad \;\,+ 8 r_{xy} p^3 - 4 (13 r_{y} p^2 - 18 r^2 + 13 r_{x} p + 23 r_{p} p r) r_{p} p + 40 r_{pp} p^2 r^2 + 28 r_{py} p^3 r\\&\quad \;\,+ 28 r_{px} p^2 r) g_{yy} - 6 (r_{y} p^2 - 3 r^2 + r_{x} p + 5 r_{p} p r) g_{yyy} p^2 + 3 g_{yyyy} p^4 r - ((3 (r_{yy} r\\&\quad \;\,- r_{y}^2) + r_{yyy} p^2) p - 3 r_{x} r_{y} + 3 r_{xxy} p + r_{xxx} + 3 r_{xy} r + 3 r_{xyy} p^2 - (17 r_{yy} p^2 - 14 r_{y} r\\&\quad \;\,+ 17 r_{xx} + 34 r_{xy} p - (113 (r_{x} + r_{y} p) + 79 r_{p} r) r_{p}) r_{p} - (39 (r_{x} + r_{y} p) + 86 r_{p} r) r_{pp} r\\&\quad \;\,+ 15 r_{ppy} p r^2 + 15 r_{ppx} r^2 + 15 r_{ppp} r^3 - (83 r_{p} p r + 27 r_{x} p + 27 r_{y} p^2 + 15 r^2) r_{py}\\&\quad \;\,+ 9 r_{pyy} p^2 r - (27 (r_{x} + r_{y} p) + 83 r_{p} r) r_{px} + 9 r_{pxx} r + 18 r_{pxy} p r) g_{x} - 4 (r_{yy} p^2\\&\quad \;\,- r_{y} r + r_{xx} + 2 r_{xy} p - (13 (r_{x} + r_{y} p) + 23 r_{p} r) r_{p} + 10 r_{pp} r^2 + 7 r_{py} p r\\&\quad \;\,+ 7 r_{px} r) g_{xx} - 18 (r_{y} p^2 - r^2 + r_{x} p + 5 r_{p} p r) g_{xxy} + 18 g_{xxyy} p^2 r - 6 (r_{x} + r_{y} p\\&\quad \;\,+ 5 r_{p} r) g_{xxx} + 12 g_{xxxy} p r + 3 g_{xxxx} r - 4 ((2 r_{yy} p^2 + 7 r_{y} r) p + 9 r_{x} r\\&\quad \;\,+ 2 r_{xx} p + 4 r_{xy} p^2 - 2 (13 r_{y} p^2 - 9 r^2 + 13 r_{x} p + 23 r_{p} p r) r_{p} + 20 r_{pp} p r^2\\&\quad \;\,+ 14 r_{py} p^2 r + 14 r_{px} p r) g_{xy} - 18 (r_{y} p^2 - 2 r^2 + r_{x} p + 5 r_{p} p r) g_{xyy} p + 12 g_{xyyy} p^3 r\\&\quad \;\,+ 2 (((5 r_{yy} p^2 - 2 r_{y} r) r_{y} - r_{yyy} p^2 r) p + (5 r_{yy} p^2 - 2 r_{y} r) r_{x} + 5 (r_{x} + r_{y} p) r_{xx}\\&\quad \;\,- 3 r_{xxy} p r - r_{xxx} r + 10 (r_{x} + r_{y} p) r_{xy} p - 3 r_{xyy} p^2 r - ((35 r_{y} p^2 + 12 r^2) r_{y}\\&\quad \;\,- 12 r_{yy} p^2 r + 35 (r_{x} + 2 r_{y} p) r_{x} - 12 r_{xx} r - 24 r_{xy} p r + (43 (r_{x} + r_{y} p)\\&\quad \;\,+ 18 r_{p} r) r_{p} r) r_{p} + (11 (r_{x} + r_{y} p) + 18 r_{p} r) r_{pp} r^2 - 3 r_{ppy} p r^3 - 3 r_{ppx} r^3 - 3 r_{ppp} r^4\\&\quad \;\,+ (17 r_{y} p^2 + 6 r^2 + 17 r_{x} p + 21 r_{p} p r) r_{py} r - 3 r_{pyy} p^2 r^2 + (17 (r_{x} + r_{y} p) + 21 r_{p} r) r_{px} r\\&\quad \;\,- 3 r_{pxx} r^2 - 6 r_{pxy} p r^2) g_{p} + ((15 r_{y} p^2 + 4 r^2) r_{y} - 4 r_{yy} p^2 r + 15 (r_{x} + 2 r_{y} p) r_{x}\\ \end{aligned}$$

$$\begin{aligned}&\quad \quad \;\;- 4 r_{xx} r - 8 r_{xy} p r + 11 (2 (r_{x} + r_{y} p) + r_{p} r) r_{p} r - 4 r_{pp} r^3 - 4 r_{py} p r^2 - 4 r_{px} r^2) g_{pp} r\\&\quad \quad \;\;- 18 (r_{x} + r_{y} p + r_{p} r) g_{ppy} p r^2 + 6 g_{ppyy} p^2 r^3 - 18 (r_{x} + r_{y} p + r_{p} r) g_{ppx} r^ + 6 g_{ppxx} r^32\\&\quad \quad \;\;+ 12 g_{ppxy} p r^3 + 2 (((15 r_{y} p^2 - r^2) r_{y} - 8 r_{yy} p^2 r) p + 3 (10 r_{y} p^2 - 3 r^2 + 5 r_{x} p) r_{x} \\&\quad \quad \;\;- 8 r_{xx} p r - 16 r_{xy} p^2 r + (74 r_{y} p^2 - 9 r^2 + 74 r_{x} p + 55 r_{p} p r) r_{p} r - 20 r_{pp} p r^3 \\&\quad \quad \;\;- 20 r_{py} p^2 r^2 - 20 r_{px} p r^2) g_{py} - 12 (3 r_{y} p^2 - r^2 + 3 r_{x} p+ 6 r_{p} p r) g_{pyy} p r + 12 g_{pyyy} p^3 r^2\\&\quad \quad \;\;+ 2 ((15 r_{y} p^2 + 8 r^2) r_{y} - 8 r_{yy} p^2 r + 15 (r_{x} + 2 r_{y} p) r_{x} - 8 r_{xx} r - 16 r_{xy} p r\\&\quad \quad \;\;+ (74 (r_{x} + r_{y} p) + 55 r_{p} r) r_{p} r - 20 r_{pp} r^3 - 20 r_{py} p r^2 - 20r_{px} r^2) g_{px} - 36 (r_{x}\\&\quad \quad \;\;+ r_{y} p + 2 r_{p} r) g_{pxx} r + 36 g_{pxxy} p r^2 + 12 g_{pxxx} r^2 - 12 (6 r_{y} p^2 - r^2 + 6 r_{x} p\\&\quad \quad \;\;+ 12 r_{p} p r) g_{pxy} r + 36 g_{pxyy} p^2 r^2) \varphi _{p}^3 - (((((7 r_{yy} p^2 + 31 r_{y} r) p + 44 r_{x}r + 7 r_{xx} p\\&\quad \quad \;\;+ 14 r_{xy} p^2 - (93 r_{y} p^2 - 100 r^2 + 93 r_{x} p + 189 r_{p} p r) r_{p} + 85 r_{pp} p r^2 + 52 r_{py} p^2 r\\&\quad \quad \;\;+ 52 r_{px} p r) g_{y} + 2 (11 r_{y} p^2 - 27 r^2 +11 r_{x} p + 61 r_{p} p r) g_{yy} p - 18 g_{yyy} p^3 r + (7 r_{yy} p^2\\&\quad \quad \;\;- 13 r_{y} r + 7 r_{xx} + 14 r_{xy} p - 3 (31 (r_{x} + r_{y} p) + 63 r_{p} r) r_{p} + 85 r_{pp} r^2 + 52 r_{py} p r\\&\quad \quad \;\;+ 52 r_{px} r) g_{x} + 2 (11 (r_{x} + r_{y} p) + 61 r_{p} r) g_{xx} - 54 g_{xxy} p r - 18 g_{xxx} r + 2 (22 r_{y} p^2\\&\quad \quad \;\;- 27 r^2 + 22 r_{x} p + 122 r_{p} p r) g_{xy} - 54 g_{xyy} p^2 r - (5 (5 r_{y} p^2 + 4 r^2) r_{y} - 14 r_{yy} p^2 r\\&\quad \quad \;\;+ 25 (r_{x} + 2 r_{y} p) r_{x} - 14 r_{xx} r - 28 r_{xy} p r + (136 (r_{x} + r_{y} p) + 121 r_{p} r) r_{p} r\\&\quad \quad \;\;- 44 r_{pp} r^3 - 38 r_{py} p r^2 - 38 r_{px} r^2) g_{p} + 2 (11 (r_{x} + r_{y} p) + 13 r_{p} r) g_{pp} r^2\\&\quad \quad \;\;- 18 g_{ppy} p r^3 - 18 g_{ppx} r^3 + 2 (44 r_{y} p^2 - 9 r^2 + 44 r_{x} p + 100 r_{p} p r) g_{py} r - 54 g_{pyy} p^2 r^2\\&\quad \quad \;\;+ 8 (11 (r_{x} + r_{y} p) + 25 r_{p} r) g_{px} r - 54 g_{pxx} r^2 - 108 g_{pxy} p r^2) \varphi _{y} + 6 (3 (g_{x} + g_{y} p)\\&\quad \quad \;\;+ 2 g_{p} r) \varphi _{yy} r^2) \varphi _{p}^2 + 5 (((5 r_{y} p^2 - 9 r^2 + 5 r_{x} p + 31 r_{p} p r) g_{y} - 9 g_{yy} p^2 r + (5 (r_{x} + r_{y} p)\\&\quad \quad \;\;+ 31 r_{p} r) g_{x} - 9 g_{xx} r - 18 g_{xy} p r + 2 (5 (r_{x} + r_{y} p) + 13 r_{p} r) g_{p} r - 3 g_{pp} r^3 - 18 g_{py} p r^2\\ \end{aligned}$$

$$\begin{aligned}&\quad \quad \;\;\;- 18 g_{px} r^2) \varphi _{p} - 9 (g_{x} + g_{y} p + g_{p} r) \varphi _{y} r) \varphi _{y}^2) - 3 (2 (3 (((2 r_{y} p^2 - 3 r^2 + r_{x} p) r_{x}\nonumber \\&\quad \quad \;\;\;+ ((r_{y} p^2 - 2 r^2) r_{y} - r_{yy} p^2 r) p - r_{xx} p r - 2 r_{xy} p^2 r + (8 r_{y} p^2 - 5 r^2 + 8 r_{x} p + 9 r_{p} p r) r_{p} r\nonumber \\&\quad \quad \;\;\;- 5 r_{pp} p r^3 - 4 r_{py} p^2 r^2 - 4 r_{px} p r^2) g_{y} - (3 r_{y} p^2 - 4 r^2 + 3 r_{x} p + 9 r_{p} p r) g_{yy} p r\nonumber \\&\quad \quad \;\;\;+ 2 g_{yyy} p^3 r^2 + ((r_{y} p^2 + r^2) r_{y} - r_{yy} p^2 r + (r_{x} + 2 r_{y} p) r_{x} - r_{xx} r - 2 r_{xy} p r\nonumber \\&\quad \quad \;\;+ (8 (r_{x} + r_{y} p) + 9 r_{p} r) r_{p} r - 5 r_{pp} r^3 - 4 r_{py} p r^2 - 4 r_{px} r^2) g_{x} - 3 (r_{x} + r_{y} p + 3 r_{p} r) g_{xx} r\nonumber \\&\quad \quad \;\;\;+ 6 g_{xxy} p r^2 + 2 g_{xxx} r^2 - 2 (3 r_{y} p^2 - 2 r^2 + 3 r_{x} p + 9 r_{p} p r) g_{xy} r + 6 g_{xyy} p^2 r^2 + ((2 r_{y} p^2\nonumber \\&\quad \quad \;\;\;+ r^2) r_{y} - r_{yy} p^2 r + 2 (r_{x} + 2 r_{y} p) r_{x} - r_{xx} r - 2 r_{xy} p r + 2 (3 (r_{x} + r_{y} p) + 2 r_{p} r) r_{p} r\nonumber \\&\quad \quad \;\;\;- 2 r_{pp} r^3 - 2 r_{py} p r^2 - 2 r_{px} r^2) g_{p} r - (r_{x} + r_{y} p + r_{p} r) g_{pp} r^3 + g_{ppy} p r^4 + g_{ppx} r^4\nonumber \\&\quad \quad \;\;\;- (6 r_{y} p^2 - r^2 + 6 r_{x} p+ 10 r_{p} p r) g_{py} r^2 + 4 g_{pyy} p^2 r^3 - 2 (3 (r_{x} + r_{y} p) + 5 r_{p} r) g_{px} r^2\nonumber \\&\quad \quad \;\;\;+ 4 g_{pxx} r^3 + 8 g_{pxy} p r^3) \varphi _{p}^2 - 5 ((2 ((2 (r_{y} p^2 - r^2 + r_{x} p) + 7 r_{p} p r) g_{y} - 3 g_{yy} p^2 \nonumber \\&\quad \quad \;\;\;+ (2 (r_{x} + r_{y} p) + 7 r_{p} r) g_{x} - 3 g_{xx} r - 6 g_{xy} p r + 2 (r_{x} + r_{y} p + 2 r_{p} r) g_{p} r) - g_{pp} r^3\nonumber \\&\quad \quad \;\;\;- 8 g_{py} p r^2 - 8 g_{px} r^2) \varphi _{p} - 3 (3 (g_{x} + g_{y} p) + 2 g_{p} r) \varphi _{y} r) \varphi _{y} r) + 5 (((6 r_{y} p^2 - 5 r^2 \nonumber \\&\quad \quad \;\;\;+ 6 r_{x} p + 14 r_{p} p r) g_{y} - 10 g_{yy} p^2 r + 2 (3 (r_{x} + r_{y} p) + 7 r_{p} r) g_{x} - 10 g_{xx} r - 20 g_{xy} p r\nonumber \\&\quad \quad \;\;\;+ 2 (2 (r_{x} + r_{y} p) + 3 r_{p} r) g_{p} r - g_{pp} r^3 - 10 g_{py} p r^2 - 10 g_{px} r^2) \varphi _{p} - 15 (2 (g_{x} + g_{y} p)\nonumber \\&\quad \quad \;\;\;+ g_{p} r) \varphi _{y} r + 3 (5 (g_{x} + g_{y} p) + 2 g_{p} r) \varphi _{pp} r^2) \varphi _{pp} r^2) \varphi _{pp} + 30 (2 (g_{x} + g_{y} p) + g_{p} r) \varphi _{ppy} \varphi _{p}^2 r^3\nonumber \\&\quad \quad \;\;\;+ 2 (2 ((6 r_{y} p^2 - 5 r^2 + 6 r_{x} p + 14 r_{p} p r) g_{y} - 10 g_{yy} p^2 r + 2 (3 (r_{x} + r_{y} p) + 7 r_{p} r) g_{x}\nonumber \\&\quad \quad \;\;\;- 10 g_{xx} r - 20 g_{xy} p r + 2 (2 (r_{x} + r_{y} p) + 3 r_{p} r) g_{p} r - g_{pp} r^3 - 10 g_{py} p r^2 - 10 g_{px} r^2) \varphi _{p}\nonumber \\&\quad \quad \;\;\;- 25 (2 (g_{x} + g_{y} p) + g_{p} r) \varphi _{y} r + 15 (5 (g_{x} + g_{y} p) + 2 g_{p} r) \varphi _{pp} r^2) \varphi _{ppp} \varphi _{p} r^2 - 3 (5 (g_{x} + g_{y} p)\nonumber \\&\quad \quad \;\;\;+ 2 g_{p} r) \varphi _{pppp} \varphi _{p}^2 r^4 - 6 (2 ((2 ((2 (r_{y} p^2 - r^2 + r_{x} p) + 7 r_{p} p r) g_{y} - 3 g_{yy} p^2 r + (2 (r_{x} + r_{y} p)\nonumber \\&\quad \quad \;\;\; + 7 r_{p} r) g_{x} - 3 g_{xx} r - 6 g_{xy} p r + 2 (r_{x} + r_{y} p + 2 r_{p} r) g_{p} r) - g_{pp} r^3 - 8 g_{py} p r^2 - 8 g_{px} r^2) \varphi _{p}\nonumber \\&\quad \quad \;\;\;- 5 (3 (g_{x} + g_{y} p) + 2 g_{p} r) \varphi _{y} r) + 25 (2 (g_{x} + g_{y} p) + g_{p} r) \varphi _{pp} r^2) \varphi _{py} \varphi _{p} r)/((g_{x} + g_{y} p\nonumber \\&\quad \quad \;\;\;- g_{p} r) \varphi _{p}^3), \end{aligned}$$

(8.17)

$$\begin{aligned} D_1&= - ((2 ((((r_{yy} r + 3 r_{y}^2 - r_{yyy} p^2) p + 3 r_{x} r_{y} - 3 r_{xxy} p - r_{xxx} + r_{xy} r - 3 r_{xyy} p^2 - (3 r_{yy} p^2\nonumber \\&\quad \;- 2 r_{y} r + 3 r_{xx} + 6 r_{xy} p + (3 (r_{x} + r_{y} p) + r_{p} r) r_{p}) r_{p} - (5 (r_{x} + r_{y} p) + 4 r_{p} r) r_{pp} r \nonumber \\&\quad \;- r_{ppy} p r^2 - r_{ppx} r^2 -r_{ppp} r^3 - 3 (r_{y} p^2 - r^2 + r_{x} p + r_{p} p r) r_{py} - r_{pyy} p^2 r - 3 (r_{x}\nonumber \\&\quad \;+ r_{y} p + r_{p} r) r_{px} - r_{pxx} r - 2 r_{pxy} p r) \varphi _{p} + (3 r_{yy} p^2 - r_{y} r + 3 r_{xx} + 6 r_{xy} p + (3 (r_{x}\nonumber \\&\quad \;+ r_{y} p) + r_{p} r) r_{p} + r_{pp} r^2 + 2 r_{py} p r + 2 r_{px} r) \varphi _{y} - 3 \varphi _{yy} r^2 + \varphi _{yyyy} p^4 - ((3 r_{y} p^2\nonumber \\&\quad \;- 4 r^2) r_{y} + 4 r_{yy} p^2 r + 3 (r_{x} + 2 r_{y} p) r_{x} + 4 r_{xx} r + 8 r_{xy} p r + 7 (2 (r_{x} + r_{y} p)\nonumber \\&\quad \;+ r_{p} r) r_{p} r + 4 r_{pp} r^3 + 4 r_{py} p r^2 + 4 r_{px} r^2) \varphi _{pp} + 6 \varphi _{ppy} r^3 - 6 (r_{x} + r_{y} p + r_{p} r) \varphi _{ppp} r^2\nonumber \\&\quad \;- \varphi _{pppp} r^4 + 2 (7 (r_{x} + r_{y} p) + 5 r_{p} r) \varphi _{py} r) \varphi _{p} r - \varphi _{yyyy} \varphi _{y} p^5)(g_{x} + g_{y} p) - (2 ((2 r_{yy} p^2\nonumber \\&\quad \;+ r_{y} r - r_{xx} - 2 r_{xy} p - (2 (r_{x} + r_{y} p) + r_{p} r) r_{p} - r_{pp} r^2 - r_{py} p r - r_{px} r) \varphi _{p}\nonumber \\&\quad \;+ (2 (r_{x} + r_{y} p) + r_{p} r) \varphi _{y} - 2 \varphi _{yyy} p^3 - 3 (r_{x} + r_{y} p + r_{p} r) \varphi _{pp} r - \varphi _{ppp} r^3 + 3 (2 r_{y} p^2\nonumber \\&\quad \;+ r^2) \varphi _{py} + 3 \varphi _{pyy} p^2 r) (5 ((\varphi _{p} r_{p} - \varphi _{y} + \varphi _{pp} r - \varphi _{py} p) r + (r_{x} - r_{y} p) \varphi _{p} + 2 \varphi _{yy} p^2\nonumber \\&\quad \;- \varphi _{py} p r) (g_{x} + g_{y} p) - 4 (g_{xx} + g_{yy} p^2 + 2 g_{xy} p) \varphi _{p} r) + 2 ((4 (g_{xx} + g_{yy} p^2\nonumber \\&\quad \;+ 2 g_{xy} p) \varphi _{y} - 5 (g_{x} + g_{y} p) \varphi _{yy} p) \varphi _{yyy} p + 6 (3 g_{xyy} p^2 + g_{xxx} + (3 g_{xxy}\nonumber \\&\quad \;+ g_{yyy} p^2) p) \varphi _{yy} \varphi _{y}) p^3 - 3 (((6 g_{xxyy} + g_{yyyy} p^2) p + 4 g_{xxxy}) p + 4 g_{xyyy} p^3 + g_{xxxx})\nonumber \\&\quad \;\times ((-p \varphi _{y} + \varphi _{p} r)^2 + \varphi _{y}^2 p^2) - 2 ((\varphi _{p} r_{p} - \varphi _{y} + \varphi _{pp} r - \varphi _{py} p) r - ((\varphi _{p} r_{y} - \varphi _{yy} p + \varphi _{py} r) p\nonumber \\&\quad \;- \varphi _{p} r_{x}))(5 (3 (g_{xx} + g_{yy} p^2 + 2 g_{xy} p) \varphi _{yy} + (g_{x} + g_{y} p) \varphi _{yyy} p) p^2 - 6 (3 g_{xyy} p^2 + g_{xxx}\nonumber \\&\quad \;+ (3 g_{xxy} + g_{yyy} p^2) p) \varphi _{p} r) - 6 (5 ((r_{x} + r_{y} p + r_{p} r) \varphi _{p} - \varphi _{y} r + \varphi _{pp} r^2)(g_{x} + g_{y} p)\nonumber \\&\quad \;- 4 (g_{xx} + g_{yy} p^2 + 2 g_{xy} p) \varphi _{p} r) (\varphi _{p} r_{yy} - \varphi _{yyy} p + 2 \varphi _{py} r_{y} + \varphi _{pyy} r) p^2 + 2 (((4 (g_{xx} + g_{yy} p^2\nonumber \\&\quad \;+ 2 g_{xy} p) \varphi _{yyy} + (g_{x} + g_{y} p) \varphi _{yyyy} p) p + 6 (3 g_{xyy} p^2 + g_{xxx} + (3 g_{xxy} + g_{yyy} p^2) p) \varphi _{yy}) p\nonumber \\&\quad - 3 (((6 g_{xxyy} + g_{yyyy} p^2) p + 4 g_{xxxy}) p + 4 g_{xyyy} p^3 + g_{xxxx}) \varphi _{y}) (\varphi _{p} r - \varphi _{y} p) p\nonumber \\&\quad \;- 15 (((\varphi _{p} r_{p} - \varphi _{y} + \varphi _{pp} r - \varphi _{py} p) r - ((\varphi _{p} r_{y} - \varphi _{yy} p + \varphi _{py} r) p- \varphi _{p} r_{x}))^2 + (4 (\varphi _{p} r_{y}\nonumber \\&\quad \;- \varphi _{yy} p + \varphi _{py} r)^2 + \varphi _{yy}^2 p^2) p^2) (g_{xx} + g_{yy} p^2 + 2 g_{xy} p) + 4 (5 ((2 r_{yy} p^2 + r_{y} r - r_{xx}\nonumber \\&\quad \;- 2 r_{xy} p - (2 (r_{x} + r_{y} p) + r_{p} r) r_{p} - r_{pp} r^2 - r_{py} p r - r_{px} r) \varphi _{p} + (2 (r_{x} + r_{y} p) + r_{p} r) \varphi _{y}\nonumber \\&\quad \;-3 \varphi _{yyy} p^3 - 3 (r_{x} + r_{y} p + r_{p} r) \varphi _{pp} r - \varphi _{ppp} r^3 + 3 (2 r_{y} p^2 + r^2) \varphi _{py} + 3 \varphi _{pyy} p^2 r) (g_{x} \nonumber \\&\quad \;+ g_{y} p) - 3 (5 ((\varphi _{p} r_{p} - \varphi _{y} + \varphi _{pp} r - \varphi _{py} p) r + (r_{x} - r_{y} p) \varphi _{p} + 2 \varphi _{yy} p^2 - \varphi _{py} p r) (g_{xx}\nonumber \\ \end{aligned}$$

(8.18)

$$\begin{aligned}&\quad \quad \;\;+ g_{yy} p^2 + 2 g_{xy} p) - 2 (3 g_{xyy} p^2 + g_{xxx} + (3 g_{xxy} + g_{yyy} p^2) p) \varphi _{p} r)) (\varphi _{p} r_{y}- \varphi _{yy} p\nonumber \\&\quad \quad \;\;+ \varphi _{py} r) p)) \varphi _{p} + ((((r_{yy} r + 3 r_{y}^2 - r_{yyy} p^2) p + 3 r_{x} r_{y} - 3 r_{xxy} p - r_{xxx} + r_{xy} r - 3 r_{xyy} p^2\nonumber \\&\quad \quad \;\;- (3 r_{yy} p^2 - 2 r_{y} r + 3 r_{xx} + 6 r_{xy} p + (3 (r_{x} + r_{y} p) + r_{p} r) r_{p}) r_{p} - (5 (r_{x} + r_{y} p)\nonumber \\&\quad \quad \;\;+ 4 r_{p} r) r_{pp} r - r_{ppy} p r^2 - r_{ppx} r^2 - r_{ppp} r^3 - 3 (r_{y} p^2 - r^2 + r_{x} p + r_{p} p r) r_{py} - r_{pyy} p^2 r\nonumber \\&\quad \quad \;\;- 3 (r_{x} + r_{y} p + r_{p} r) r_{px} - r_{pxx} r - 2 r_{pxy} p r) \varphi _{p} + (3 r_{yy} p^2 - r_{y} r + 3 r_{xx} + 6 r_{xy} p\nonumber \\&\quad \quad \;\;+ (3 (r_{x} + r_{y} p) + r_{p} r) r_{p} + r_{pp} r^2 + 2 r_{py} p r + 2 r_{px} r) \varphi _{y} - 3 \varphi _{yy} r^2 + \varphi _{yyyy} p^4\nonumber \\&\quad \quad \;\;- ((3 r_{y} p^2 - 4 r^2) r_{y} + 4 r_{yy} p^2 r + 3 (r_{x} + 2 r_{y} p) r_{x} + 4 r_{xx} r + 8 r_{xy} p r + 7 (2 (r_{x} + r_{y} p)\nonumber \\&\quad \quad \;\;+ r_{p} r) r_{p} r + 4 r_{pp} r^3 + 4 r_{py} p r^2 + 4 r_{px} r^2) \varphi _{pp} + 6 \varphi _{ppy} r^3 - 6 (r_{x} + r_{y} p + r_{p} r) \varphi _{ppp} r^2\nonumber \\&\quad \quad \;\;- \varphi _{pppp} r^4 + 2 (7 (r_{x} + r_{y} p) + 5 r_{p} r) \varphi _{py} r) g_{p} - 4 (((r_{p}^2 - 2 r_{y}) r_{p} - (r_{xy} + r_{yy} p)\nonumber \\&\quad \quad \;\;+ 2 (r_{x} + r_{y} p + 2 r_{p} r) r_{pp} + r_{ppy} p r + r_{ppx} r + r_{ppp} r^2 + 3 (r_{p} p - r) r_{py} + r_{pyy} p^2 + 3 r_{px} r_{p}\nonumber \\&\quad \quad \;\;+ r_{pxx} + 2 r_{pxy} p) \varphi _{p} - ((r_{p}^2 - r_{y} + r_{pp} r + 2 r_{py} p + 2 r_{px}) \varphi _{y} - 3 \varphi _{yy} r) + (r_{yy} p^2 - 4 r_{y} r\nonumber \\&\quad \quad \;\; + r_{xx} + 2 r_{xy} p + (5 (r_{x} + r_{y} p) + 7 r_{p} r) r_{p} + 4 r_{pp} r^2 + 4 r_{py} p r + 4 r_{px} r) \varphi _{pp} - 6 \varphi _{ppy} r^2\nonumber \\&\quad \quad \;\; + 3 (r_{x} + r_{y} p + 2 r_{p} r) \varphi _{ppp} r + \varphi _{pppp} r^3 - 5 (r_{x} + r_{y} p + 2 r_{p} r) \varphi _{py}) (g_{x} + g_{y} p)) \varphi _{p}^2 r^2\nonumber \\&\quad \quad \;\;- ((60 (3 ((\varphi _{p} r_{p} - \varphi _{y} + \varphi _{pp} r - \varphi _{py} p) r + (r_{x} - r_{y} p) \varphi _{p} + 2 \varphi _{yy} p^2 - \varphi _{py} p r) g_{p} p\nonumber \\&\quad \quad \;\;- ((g_{x} + 2 g_{y} p + 2 g_{py} p^2 + 2 g_{px} p) \varphi _{p} r - 3 (g_{x} + g_{y} p) \varphi _{y} p)) (\varphi _{p} r_{y} - \varphi _{yy} p + \varphi _{py} r)^2\nonumber \\&\quad \quad \;\;- (18 (g_{xy} p + g_{xx} - g_{pyy} p^3 - g_{pxx} p - 2 g_{pxy} p^2) \varphi _{yy} \varphi _{y} + (10 \varphi _{yyy} \varphi _{yy} g_{p} p^2 - 8 \varphi _{yyy} \varphi _{y} g_{px} p\nonumber \\&\quad \quad \;\;- 8 \varphi _{yyy} \varphi _{y} g_{py} p^2 + 4 \varphi _{yyy} \varphi _{y} g_{x} + 30 \varphi _{yy}^2 g_{px} p + 30 \varphi _{yy}^2 g_{py} p^2 - 15 \varphi _{yy}^2 g_{x}) p) \varphi _{y} p^2) p\nonumber \\&\quad \quad \;\;+ (15 ((\varphi _{p} r_{p} - \varphi _{y} + \varphi _{pp} r - \varphi _{py} p) r - ((\varphi _{p} r_{y} - \varphi _{yy} p + \varphi _{py} r) p - \varphi _{p} r_{x}))^3 + (120 (\varphi _{p} r_{y}\nonumber \\&\quad \quad \;\;- \varphi _{yy} p + \varphi _{py} r)^3 + (\varphi _{yyyy} \varphi _{y}^2 + 15 \varphi _{yy}^3) p^3) p^3) g_{p} - 2 (3 (g_{xxy} + g_{yyy} p^2 + 2 g_{xyy} p) + 2 g_{pyyy} p^3\nonumber \\&\quad \quad \;\;+ 6 g_{pxxy} p + 2 g_{pxxx} + 6 g_{pxyy} p^2) (- p \varphi _{y} + \varphi _{p} r)^3 + 2 (3 (g_{xxx} + 2 g_{xxy} p + g_{xyy} p^2)\nonumber \\&\quad \quad \;\;- 2 g_{pyyy} p^4 - 6 g_{pxxy} p^2 - 2 g_{pxxx} p - 6 g_{pxyy} p^3) \varphi _{y}^3 p^2 - 6 ((5 (r_{x} + r_{y} p + r_{p} r) g_{p} p\nonumber \\&\quad \quad \;\;- 2 (g_{x} + 2 g_{y} p) r - 4 g_{py} p^2 r - 4 g_{px} p r) \varphi _{p} + 5 (g_{x} + g_{y} p - g_{p} r) \varphi _{y} p + 5 \varphi _{pp} g_{p} p r^2) (\varphi _{p} r_{yy}\nonumber \\&\quad \quad \;\;- \varphi _{yyy} p + 2 \varphi _{py} r_{y} + \varphi _{pyy} r) \varphi _{p} p r - ((6 (g_{xx} - 5 g_{yy} p^2 -4 g_{xy} p - 6 g_{pyy} p^3 - 6 g_{pxx} p\nonumber \\&\quad \quad \;\;- 12 g_{pxy} p^2) \varphi _{yy} \varphi _{y} - (2 ((g_{x} + 5 g_{y} p + 8 g_{py} p^2 + 8 g_{px} p) \varphi _{y} - 5 \varphi _{yy} g_{p} p^2) \varphi _{yyy} \nonumber \\&\quad \quad \;\;+ 2 \varphi _{yyyy} \varphi _{y} g_{p} p^2 - 30 \varphi _{yy}^2 g_{px} p - 30 \varphi _{yy}^2 g_{py} p^2 - 30 \varphi _{yy}^2 g_{x} - 45 \varphi _{yy}^2 g_{y} p) p) p + 6 (2 (g_{pyyy} p^4\nonumber \\&\quad \quad \;\; - g_{xxx} + 3 g_{pxxy} p^2 + g_{pxxx} p + 3 g_{pxyy} p^3) - (3 g_{xxy} - g_{yyy} p^2) p) \varphi _{y}^2) (\varphi _{p} r - \varphi _{y} p) p\nonumber \\&\quad \quad \;\;+ (6 (g_{xxx} - 2 g_{yyy} p^3 - 3 g_{xyy} p^2 - 2 g_{pyyy} p^4 - 6 g_{pxxy} p^2 - 2 g_{pxxx} p - 6 g_{pxyy} p^3) \varphi _{y}\nonumber \\&\quad \quad \;\;+ (6 (2 g_{xx} + 5 g_{yy} p^2 + 7 g_{xy} p + 3 g_{pyy} p^3 + 3 g_{pxx} p + 6 g_{pxy} p^2) \varphi _{yy} + (\varphi _{yyyy} g_{p} p^2\nonumber \\&\quad \quad \;\;+ 8 \varphi _{yyy} g_{px} p + 8 \varphi _{yyy} g_{py} p^2 + 6 \varphi _{yyy} g_{x} + 10 \varphi _{yyy} g_{y} p) p) p) ( - p \varphi _{y} + \varphi _{p} r)^2 + (9 (2 (g_{xy}\nonumber \\&\quad \quad \;\;+ g_{yy} p + g_{pyy} p^2 + g_{pxx} + 2 g_{pxy} p) \varphi _{p}^2 r^2 + 5 (2 (g_{x} + g_{y} p) \varphi _{y} + \varphi _{yy} g_{p} p^2) \varphi _{yy} p^2)\nonumber \\&\quad \quad \;\;- 10 (3 ((g_{x} + 2 g_{y} p + 2 g_{py} p^2 + 2 g_{px} p) \varphi _{yy} p + (g_{xx} + g_{yy} p^2 + 2 g_{xy} p) \varphi _{y}) + \varphi _{yyy} g_{p} p^3) \varphi _{p} r\nonumber \\ \end{aligned}$$

$$\begin{aligned} D_0&= - (((((r_{yy} r + 3 r_{y}^2 - r_{yyy} p^2) p + 3 r_{x} r_{y} - 3 r_{xxy} p - r_{xxx} + r_{xy} r - 3 r_{xyy} p^2 - (3 r_{yy} p^2\nonumber \\&\quad - 2 r_{y} r + 3 r_{xx} + 6 r_{xy} p + (3 (r_{x} + r_{y} p) + r_{p} r) r_{p}) r_{p} - (5 (r_{x} + r_{y} p) + 4 r_{p} r) r_{pp} r \nonumber \\&\quad - r_{ppy} p r^2 - r_{ppx} r^2 - r_{ppp} r^3 - 3 (r_{y} p^2 - r^2 + r_{x} p + r_{p} p r) r_{py} - r_{pyy} p^2 r - 3 (r_{x} + r_{y} p\nonumber \\&\quad + r_{p} r) r_{px} - r_{pxx} r - 2 r_{pxy} p r) \varphi _{p} + (3 r_{yy} p^2 - r_{y}r + 3 r_{xx} + 6 r_{xy} p + (3 (r_{x} + r_{y} p)\nonumber \\&\quad + r_{p} r) r_{p} + r_{pp} r^2 + 2 r_{py} p r + 2 r_{px} r) \varphi _{y} - 3 \varphi _{yy} r^2 + \varphi _{yyyy} p^4 - ((3 r_{y} p^2 - 4 r^2) r_{y}\nonumber \\&\quad + 4 r_{yy} p^2 r + 3 (r_{x} + 2 r_{y} p) r_{x} + 4 r_{xx} r + 8 r_{xy} p r + 7 (2 (r_{x} + r_{y} p) + r_{p} r) r_{p} r + 4 r_{pp} r^3\nonumber \\&\quad + 4 r_{py} p r^2 + 4 r_{px} r^2) \varphi _{pp} + 6 \varphi _{ppy} r^3 - 6 (r_{x} + r_{y} p + r_{p} r) \varphi _{ppp} r^2 - \varphi _{pppp} r^4 + 2 (7 (r_{x}\nonumber \\&\quad + r_{y} p) + 5 r_{p} r) \varphi _{py} r) \varphi _{p}^2 r^2 - (15 ((\varphi _{p} r_{p} - \varphi _{y} + \varphi _{pp} r - \varphi _{py} p) r - ((\varphi _{p} r_{y} - \varphi _{yy} p\nonumber \\&\quad + \varphi _{py} r) p - \varphi _{p} r_{x}))^3 + (120 (\varphi _{p} r_{y} - \varphi _{yy} p + \varphi _{py} r)^3 + (\varphi _{yyyy} \varphi _{y}^2 + 15 \varphi _{yy}^3) p^3) p^3)\nonumber \\&\quad \times (g_{x} + g_{y} p) - ((2 (5 (2 ((2 r_{yy} p^2 + r_{y} r - r_{xx} - 2 r_{xy} p - (2 (r_{x} + r_{y} p) + r_{p} r) r_{p}\nonumber \\&\quad - r_{pp} r^2 - r_{py} p r - r_{px} r) \varphi _{p} + (2 (r_{x} + r_{y} p) + r_{p} r) \varphi _{y} - 2 \varphi _{yyy} p^3 - 3 (r_{x} + r_{y} p \nonumber \\&\quad + r_{p} r) \varphi _{pp} r - \varphi _{ppp} r^3 + 3 (2 r_{y} p^2 + r^2) \varphi _{py} + 3 \varphi _{pyy} p^2 r) \varphi _{p} r + 9 ((\varphi _{p} r_{p} - \varphi _{y} + \varphi _{pp} r\nonumber \\&\quad - \varphi _{py} p) r - ((\varphi _{p} r_{y} - \varphi _{yy} p + \varphi _{py} r) p - \varphi _{p} r_{x}))^2 - (2 \varphi _{yyy} \varphi _{y} - 9 \varphi _{yy}^2) p^4)(g_{x} + g_{y} p)\nonumber \\&\quad - 2 ((5 (3 (g_{xx} + g_{yy} p^2 + 2 g_{xy} p) \varphi _{yy} + (g_{x} + g_{y} p) \varphi _{yyy} p) p - 6 (3 g_{xyy} p^2 + g_{xxx}\nonumber \\&\quad + (3 g_{xxy} + g_{yyy} p^2) p) \varphi _{y}) (\varphi _{p} r - \varphi _{y} p) p - 3 ((( - p \varphi _{y} + \varphi _{p} r)^2 + \varphi _{y}^2 p^2) (3 g_{xyy} p^2\nonumber \\&\quad + g_{xxx} + (3 g_{xxy} + g_{yyy} p^2) p) - 5 (g_{xx} + g_{yy} p^2+ 2 g_{xy} p) \varphi _{yy} \varphi _{y} p^3) + 15 ((\varphi _{p} r_{p}\nonumber \\&\quad - \varphi _{y} + \varphi _{pp} r - \varphi _{py} p) r - ((\varphi _{p} r_{y} - \varphi _{yy} p + \varphi _{py} r) p - \varphi _{p} r_{x})) ((g_{xx} + g_{yy} p^2 + 2 g_{xy} p) \varphi _{p} r\nonumber \\&\quad - 3 (g_{x} + g_{y} p) \varphi _{yy} p^2))) (\varphi _{p} r_{y} - \varphi _{yy} p + \varphi _{py} r) - (((15 (g_{xx} + g_{yy} p^2 + 2 g_{xy} p) \varphi _{yy}^2\nonumber \\&\quad - 2 (g_{x} + g_{y} p) \varphi _{yyyy} \varphi _{y} p - 2 (4 (g_{xx} + g_{yy} p^2 + 2 g_{xy} p) \varphi _{y} - 5 (g_{x} + g_{y} p) \varphi _{yy} p) \varphi _{yyy}) p\nonumber \\&\quad - 12 (3 g_{xyy} p^2 + g_{xxx} + (3 g_{xxy} + g_{yyy} p^2) p) \varphi _{yy} \varphi _{y}) p + 3 (((6 g_{xxyy} + g_{yyyy} p^2) p\nonumber \\ \end{aligned}$$

(8.19)

$$\begin{aligned}&\quad \quad \;\;+ 4 g_{xxxy}) p + 4 g_{xyyy} p^3 + g_{xxxx}) \varphi _{y}^2) (\varphi _{p} r - \varphi _{y} p) p) p + 2 ((2 r_{yy} p^2 + r_{y} r - r_{xx}\nonumber \\&\quad \quad \;\;- 2 r_{xy} p - (2 (r_{x} + r_{y} p) + r_{p} r) r_{p} - r_{pp} r^2 - r_{py} p r - r_{px} r) \varphi _{p} + (2 (r_{x} + r_{y} p)\nonumber \\&\quad \quad \;\;+ r_{p} r) \varphi _{y} - 2 \varphi _{yyy} p^3 - 3 (r_{x} + r_{y} p + r_{p} r) \varphi _{pp} r - \varphi _{ppp} r^3 + 3 (2 r_{y} p^2 + r^2) \varphi _{py}\nonumber \\&\quad \quad \;\;+ 3 \varphi _{pyy} p^2 r) (5 ((\varphi _{p} r_{p} - \varphi _{y} + \varphi _{pp} r - \varphi _{py} p) r - ((\varphi _{p} r_{y} - \varphi _{yy} p + \varphi _{py} r) p\nonumber \\&\quad \quad \;\;- \varphi _{p} r_{x})) (g_{x} + g_{y} p) - (2 (g_{xx} + g_{yy} p^2 + 2 g_{xy} p) \varphi _{p} r - 5 (g_{x} + g_{y} p) \varphi _{yy} p^2)) \varphi _{p} r\nonumber \\&\quad \quad \;\;+ (3 (2 (3 g_{xyy} p^2 + g_{xxx} + (3 g_{xxy} + g_{yyy} p^2) p) \varphi _{y} - 5 (g_{xx} + g_{yy} p^2 + 2 g_{xy} p) \varphi _{yy} p) \varphi _{yy}\nonumber \\&\quad \quad \;\;+ 2 (2 (g_{xx} + g_{yy} p^2 + 2 g_{xy} p) \varphi _{y} - 5 (g_{x} + g_{y} p) \varphi _{yy} p) \varphi _{yyy} p) \varphi _{y} p^4 - (((6 g_{xxyy} \nonumber \\&\quad \quad \;\;+ g_{yyyy} p^2) p + 4 g_{xxxy}) p + 4 g_{xyyy} p^3+ g_{xxxx}) (\varphi _{p}^2 r^2 - 3 \varphi _{p} \varphi _{y} p r + 3 \varphi _{y}^2 p^2) \varphi _{p} r\nonumber \\&\quad \quad \;\;+ (((4 (g_{xx} + g_{yy} p^2+ 2 g_{xy} p) \varphi _{yyy} + (g_{x} + g_{y} p) \varphi _{yyyy} p) p + 6 (3 g_{xyy} p^2 + g_{xxx}\nonumber \\&\quad \quad \;\;+ (3 g_{xxy} + g_{yyy} p^2) p) \varphi _{yy}) p - 3 (((6 g_{xxyy} + g_{yyyy} p^2) p + 4 g_{xxxy}) p + 4 g_{xyyy} p^3\nonumber \\&\quad \quad \;\;+ g_{xxxx}) \varphi _{y}) ( - p \varphi _{y} + \varphi _{p} r)^2 p + 60 (3 ((\varphi _{p} r_{p} - \varphi _{y} + \varphi _{pp} r - \varphi _{py} p) r\nonumber \\&\quad \quad \;\;+ (r_{x} - r_{y} p) \varphi _{p} + 2 \varphi _{yy} p^2 - \varphi _{py} p r) (g_{x} + g_{y} p) - (g_{xx} + g_{yy} p^2 + 2 g_{xy} p) \varphi _{p} r)\nonumber \\&\quad \quad \;\;\times (\varphi _{p} r_{y} - \varphi _{yy} p + \varphi _{py} r)^2 p^2 - (5 (2 (3 (g_{xx} + g_{yy} p^2 + 2 g_{xy} p) \varphi _{yy} + (g_{x} + g_{y} p) \varphi _{yyy} p) \varphi _{p} r\nonumber \\&\quad \quad \;\;- 9 (g_{x} + g_{y} p) \varphi _{yy}^2 p^2) p^2 - 6 ((3 g_{xxy} + g_{yyy} p^2) p + g_{xxx} + 3 g_{xyy} p^2) \varphi _{p}^2 r^2\nonumber \\&\quad \quad \;\;+ 15 ((\varphi _{p} r_{p} - \varphi _{y} + \varphi _{pp} r - \varphi _{py} p) r - ((\varphi _{p} r_{y} - \varphi _{yy} p + \varphi _{py} r) p - \varphi _{p} r_{x})) ((g_{xx} \nonumber \\&\quad \quad \;\;+ g_{yy} p^2 + 2 g_{xy} p) \varphi _{p} r - 3 (g_{x} + g_{y} p) \varphi _{yy} p^2))((\varphi _{p} r_{p} - \varphi _{y} + \varphi _{pp} r - \varphi _{py} p) r \nonumber \\&\quad \quad \;\;-((\varphi _{p} r_{y} - \varphi _{yy} p + \varphi _{py} r) p - \varphi _{p} r_{x})) - 6 (((5 (r_{x} + r_{y} p + r_{p} r) g_{y} - 2 g_{yy} p r) p \nonumber \\&\quad \quad \;\;+ 5 (r_{x} + r_{y} p + r_{p} r) g_{x} - 2 g_{xx} r - 4 g_{xy} p r) \varphi _{p} - 5 (g_{x} + g_{y} p) \varphi _{y} r + 5 (g_{x} \nonumber \\&\quad \quad \;\;+ g_{y} p) \varphi _{pp} r^2)(\varphi _{p} r_{yy} - \varphi _{yyy} p + 2 \varphi _{py} r_{y} + \varphi _{pyy} r) \varphi _{p} p^2 r))/((g_{x} + g_{y} p - g_{p} r) \varphi _{p}^3) . \end{aligned}$$

$$\begin{aligned} \lambda _{4}&= - 126 B_{0y} A_{2} + 399 B_{1y} A_{2} r + 35 B_{1y} B_{1} + 5 B_{1y} \lambda _{1} - 675 C_{2y} + 2025 C_{3y} r - 4050 C_{4y} r^2,\\ \lambda _{5}&=- 2551500 B_{0x} A_{2} + 6075000 B_{0y} +6189750 B_{1p} A_{2} r^2 - 270000 B_{1p} B_{0} + 1001250 B_{1p} B_{1} r \\&\quad -180000 B_{1p} \lambda _{1} r + 8079750 B_{1x} A_{2} r + 708750 B_{1x} B_{1}+101250 B_{1x} \lambda _{1} - 9112500 B_{1y} r \\&\quad - 13668750 C_{2x} + 41006250 C_{3x} r- 82012500 C_{4x} r^2 + 7593750 \lambda _{1y} r + 5817258 A_{2}^3 r^3 \\&\quad + 5017950 A_{2}^2 B_{0} r- 1030104 A_{2}^2 B_{1} r^2 - 2949318 A_{2}^2 \lambda _{1} r^2 + 768150 A_{2} B_{0} B_{1} + 112050 A_{2} B_{0} \lambda _{1} \\&\quad - 700812 A_{2} B_{1}^2 r - 416988 A_{2} B_{1} \lambda _{1} r - 11481750 A_{2} C_{1} + 35690625 A_{2} C_{2} r - 35008875 A_{2} C_{3} r^2 \\&\quad + 10800000 A_{2} C_{4} r^3 + 77247 A_{2} \lambda _{1}^2 r - 8343000 A_{2} \lambda _{2} r^3 +8490 A_{2} \lambda _{3} r^2 + 1451250 B_{0} C_{3} \\&\quad - 6615000 B_{0} C_{4} r + 344250 B_{0} \lambda _{2} r- 330 B_{0} \lambda _{3} - 47250 B_{1}^3 - 180 B_{1}^2 \lambda _{1} + 2683125 B_{1} C_{2} \\&\quad - 4300875 B_{1} C_{3} r + 6234750 B_{1} C_{4} r^2 + 4455 B_{1} \lambda _{1}^2 - 1208250 B_{1} \lambda _{2} r^2 +1118 B_{1} \lambda _{3} r \\&\quad - 50625 C_{2} \lambda _{1} - 85500 C_{3} \lambda _{1} r + 409500 C_{4} \lambda _{1} r^2 -82012500 D_{3} + 287043750 D_{4} r \\&\quad -615093750 D_{5} r^2 + 1025156250 D_{6} r^3 -1435218750 D_{7} r^4 - 7245 \lambda _{1}^3 + 47250 \lambda _{1} \lambda _{2} r^2 \\&\quad - 319 \lambda _{1} \lambda _{3} r,\\ \lambda _{6}&=425250 A_{2x} r^2 + 20250 B_{0x} - 76500 B_{1p} r^2 + 20250 B_{1x} r - 178578 A_{2}^2 r^3 - 4050 A_{2} B_{0} r \\&\quad -59346 A_{2} B_{1} r^2 + 17388 A_{2} \lambda _{1} r^2 - 4500 B_{0} B_{1} + 882 B_{1}^2 r + 558 B_{1} \lambda _{1} r + 91125 C_{1} \\&\quad - 50625 C_{2} r - 292500 C_{3} r^2 + 758250 C_{4} r^3 -837 \lambda _{1}^2 r + 70200 \lambda _{2} r^3 - 44 \lambda _{3} r^2,\\ \lambda _{7}&= - 45 C_{4pyy} + 18 C_{4yy} A_{2} +1070 D_{7yy} r,\\ \lambda _{8}&= - 9499200 B_{1y} D_{7} + 34316235 C_{4py} A_{2} - 15393966 C_{4y} A_{2}^2 + 30356800 C_{4y} C_{4} \\&\quad + 399120 C_{4y} \lambda _{2} +135792000 D_{6py} + 13893840 D_{6y} A_{2} + 442980000 D_{7xy} \\&\quad -37037360 D_{7y} B_{1} - 14850920 D_{7y} \lambda _{1},\\ \lambda _{9}&= - 44208000 B_{1p} C_{4y} + 2922750 B_{1y} C_{4p} - 5527620 B_{1y} A_{2}^3 - 25323900 B_{1y} A_{2} C_{4} \\&\quad - 5131332 B_{1y} A_{2} \lambda _{2} - 112873500 B_{1y} D_{6} -139725000 C_{3py} A_{2} - 321367500 C_{3ppy} \\&\quad - 19282050 C_{3y} A_{2}^2- 169765875 C_{3y} C_{4} + 4610925 C_{3y} \lambda _{2} - 419175000 C_{4pxy} \\&\quad +34838100 C_{4py} B_{1} + 36553950 C_{4py} \lambda _{1} + 167670000 C_{4xy} A_{2} - 838350000 C_{4yy} \\&\quad - 31378860 C_{4y} A_{2} B_{1} + 13106880 C_{4y} A_{2} \lambda _{1} - 203670000 C_{4y} C_{3} - 21308 C_{4y} \lambda _{3} \\&\quad - 1173690000 D_{5py} - 142519500 D_{5y} A_{2} + 42476400 D_{6y} B_{1} + 362539800 D_{6y} \lambda _{1} \\&\quad +110745000 D_{7y} B_{0},\\ \end{aligned}$$

$$\begin{aligned} \lambda _{10}&=153180000 D_{7pp} + 1528216550 D_{7p} A_{2} - 165584497 A_{2}^2 D_{7} - 3309109750 C_{4} D_{7} \\&\quad - 1411142550 D_{7} \lambda _{2},\\ \lambda _{11}&=- 732623400000 B_{1p} D_{7} - 736805700 C_{4p} A_{2}^2 + 332845425000 C_{4p} C_{4} \\&\quad +119274660000 C_{4p} \lambda _{2}- 103396500000 D_{6pp} - 3616060860000 D_{6p} A_{2} \\&\quad + 42277680000 D_{7p} B_{1}- 9400590000 D_{7p} \lambda _{1}- 5414139967500 D_{7x} A_{2} \\&\quad + 59005320750000 D_{7y} + 21830228928 A_{2}^5+ 55914404100 A_{2}^3 C_{4} \\&\quad + 1469327400 A_{2}^3 \lambda _{2} - 693906693300 A_{2}^2 D_{6}+ 243706787100 A_{2} B_{1} D_{7} \\&\quad - 466375751375 A_{2} C_{4}^2- 18117545100 A_{2} C_{4} \lambda _{2} + 174420916200 A_{2} D_{7} \lambda _{1} \\&\quad - 4774589325 A_{2} \lambda _{2}^2 - 1302509025000 C_{3} D_{7} \\&\quad + 1018922512500 C_{4} D_{6} +631870402500 D_{6} \lambda _{2} + 384816000 D_{7} \lambda _{3},\\ \lambda _{12}&=41520448125000 B_{1p} C_{4p} + 11335822803000 B_{1p} A_{2}^3 +44298079177500 B_{1p} A_{2} C_{4}\\&\quad - 15540361366500 B_{1p} A_{2} \lambda _{2} +269197445250000 B_{1p} D_{6} - 66363786000000 B_{1x} D_{7}\\&\quad + 390488025318750 C_{3pp} A_{2} + 171289115032500 C_{3p} A_{2}^2 - 41512646812500 C_{3p} C_{4}\\&\quad - 28291587862500 C_{3p} \lambda _{2} + 239741796768750 C_{4px} A_{2} -2432839303125000 C_{4py}\\&\quad - 13794922100250 C_{4p} A_{2} B_{1} - 12976926306750 C_{4p} A_{2} \lambda _{1} + 81266272312500 C_{4p} C_{3}\\&\quad - 31190553000 C_{4p} \lambda _{3}- 107546094967500 C_{4x} A_{2}^2 + 212080194000000 C_{4x} C_{4}\\&\quad +2788352100000 C_{4x} \lambda _{2} + 2071537656975000 C_{4y} A_{2} + 1426130179425000 D_{5p} A_{2}\\&\quad + 948676860000000 D_{6px} - 109624881600000 D_{6p} B_{1} +12190726800000 D_{6p} \lambda _{1}\\&\quad + 97065839700000 D_{6x} A_{2} + 2898092115000000 D_{6y} - 23781195000000 D_{7p} B_{0}\\&\quad + 1547384512500000 D_{7xx} -258752256300000 D_{7x} B_{1} - 103752239850000 D_{7x} \lambda _{1}\\&\quad + 6244921910160 A_{2}^4 B_{1} - 3019279445280 A_{2}^4 \lambda _{1} + 33023049583500 A_{2}^3 C_{3}\\&\quad + 9643894779 A_{2}^3 \lambda _{3} + 35671105518750 A_{2}^2 B_{1} C_{4} - 2509140526740 A_{2}^2 B_{1} \lambda _{2}\\&\quad -30800212453125 A_{2}^2 C_{4} \lambda _{1} + 273509864088750 A_{2}^2 D_{5} - 4854310235595 A_{2}^2 \lambda _{1} \lambda _{2}\\&\quad + 20742120360000 A_{2} B_{0} D_{7} - 15693179841000 A_{2} B_{1} D_{6} +267978840778125 A_{2} C_{3} C_{4}\\&\quad - 36979813105875 A_{2} C_{3} \lambda _{2} + 76302715165 A_{2} C_{4} \lambda _{3} - 329871306019500 A_{2} D_{6} \lambda _{1}\\&\quad + 11340571245 A_{2} \lambda _{2} \lambda _{3} + 8332595388000 B_{1}^2 D_{7} + 3717891292500 B_{1} C_{4}^2\\&\quad + 8141834434500 B_{1} C_{4} \lambda _{2} +14847726222000 B_{1} D_{7} \lambda _{1} + 647928298800 B_{1} \lambda _{2}^2\\&\quad + 33948983250000 C_{2} D_{7}+ 711226564875000 C_{3} D_{6} - 19481835015000 C_{4}^2 \lambda _{1}\\&\quad - 150189354562500 C_{4} D_{5} - 6178708431000 C_{4} \lambda _{1} \lambda _{2} - 136336459162500 D_{5} \lambda _{2}\\&\quad - 118965798000 D_{6} \lambda _{3} - 4360101858000 D_{7} \lambda _{1}^2 - 197255930400 \lambda _{1} \lambda _{2}^2,\\ \end{aligned}$$

$$\begin{aligned} \lambda _{13}&=- 1491210000000 B_{1p}^2 A_{2} - 7063098750000 B_{1p} C_{3p} - 4476060000000 B_{1p} C_{4x}\\&\quad - 830078280000 B_{1p} A_{2}^2 B_{1} - 241173247500 B_{1p} A_{2}^2 \lambda _{1} - 7866841500000 B_{1p} A_{2} C_{3}\\&\quad - 144765000 B_{1p} A_{2} \lambda _{3} + 335570850000 B_{1p} B_{1} C_{4} -47953620000 B_{1p} B_{1} \lambda _{2}\\&\quad + 240163481250 B_{1p} C_{4} \lambda _{1} -15311733750000 B_{1p} D_{5} + 132158891250 B_{1p} \lambda _{1} \lambda _{2}\\&\quad + 295928437500 B_{1x} C_{4p} - 559671525000 B_{1x} A_{2}^3 - 2564044875000 B_{1x} A_{2} C_{4}\\&\quad - 519547365000 B_{1x} A_{2} \lambda _{2} - 11428441875000 B_{1x} D_{6} +7698413137500 B_{1y} A_{2}^2\\&\quad - 5520124687500 B_{1y} C_{4} - 840956175000 B_{1y} \lambda _{2} - 14147156250000 C_{3px} A_{2}\\&\quad + 5658862500000 C_{3py} -32538459375000 C_{3ppx} + 3470769000000 C_{3pp} B_{1}\\&\quad + 1172642062500 C_{3pp} \lambda _{1} - 311866200000 C_{3p} A_{2} B_{1} - 1368269212500 C_{3p} A_{2} \lambda _{1}\\&\quad - 19119619687500 C_{3p} C_{3} + 1364647500 C_{3p} \lambda _{3} - 1952307562500 C_{3x} A_{2}^2\\&\quad - 17188794843750 C_{3x} C_{4} + 466856156250 C_{3x} \lambda _{2} +28718727187500 C_{3y} A_{2}\\&\quad - 21220734375000 C_{4pxx} + 3527357625000 C_{4px} B_{1} + 3701087437500 C_{4px} \lambda _{1}\\&\quad - 559826437500 C_{4p} A_{2} B_{0} - 70281067500 C_{4p} B_{1}^2 - 393602895000 C_{4p} B_{1} \lambda _{1}\\&\quad - 2419710468750 C_{4p} C_{2} - 12753551250 C_{4p} \lambda _{1}^2 - 255509943750000 C_{4xy}\\&\quad +8488293750000 C_{4xx} A_{2} - 3177109575000 C_{4x} A_{2} B_{1} + 1327071600000 C_{4x} A_{2} \lambda _{1}\\&\quad - 20621587500000 C_{4x} C_{3} - 2157435000 C_{4x} \lambda _{3} +20275130250000 C_{4y} B_{1}\\&\quad - 2347744500000 C_{4y} \lambda _{1} - 118836112500000 D_{5px} + 12675852000000 D_{5p} B_{1}\\&\quad + 4282692750000 D_{5p} \lambda _{1} -14430099375000 D_{5x} A_{2} - 155618718750000 D_{5y}\\&\quad + 6874470000000 D_{6p} B_{0} + 4300735500000 D_{6x} B_{1} + 36707154750000 D_{6x} \lambda _{1}\\&\quad +11212931250000 D_{7x} B_{0} + 1697795437500 \lambda _{1y} A_{2}^2 + 1486134843750 \lambda _{1y} C_{4}\\&\quad - 3653971256250 \lambda _{1y} \lambda _{2} + 3674362500 \lambda _{3y} A_{2} -261180045000 A_{2}^4 B_{0}\\&\quad - 97952473800 A_{2}^3 B_{1}^2 - 43228744200 A_{2}^3 B_{1} \lambda _{1} - 1399178812500 A_{2}^3 C_{2}\\&\quad + 1457821800 A_{2}^3 \lambda _{1}^2 - 132059362500 A_{2}^2 B_{0} C_{4} - 74953107000 A_{2}^2 B_{0} \lambda _{2}\\&\quad - 2267471272500 A_{2}^2 B_{1} C_{3} -316216440 A_{2}^2 B_{1} \lambda _{3} - 493167588750 A_{2}^2 C_{3} \lambda _{1}\\&\quad - 5177859187500 A_{2}^2 D_{4} + 199095570 A_{2}^2 \lambda _{1} \lambda _{3} - 6636907125000 A_{2} B_{0} D_{6}\\&\quad + 26696385000 A_{2} B_{1}^2 C_{4} - 55382178600 A_{2} B_{1}^2 \lambda _{2} + 432136890000 A_{2} B_{1} C_{4} \lambda _{1}\\&\quad -108748575000 A_{2} B_{1} D_{5} + 131579316000 A_{2} B_{1} \lambda _{1} \lambda _{2} - 590915250000 A_{2} C_{2} C_{4}\\ \end{aligned}$$

$$\begin{aligned}&\quad \quad - 42600937500 A_{2} C_{2} \lambda _{2} - 10497944250000 A_{2} C_{3}^2 - 1530479250 A_{2} C_{3} \lambda _{3}\\&\quad \quad - 319094403750 A_{2} C_{4} \lambda _{1}^2 - 5795167275000 A_{2} D_{5} \lambda _{1} - 24783962850 A_{2} \lambda _{1}^2 \lambda _{2}\\&\quad \quad - 615278 A_{2} \lambda _{3}^2 - 579299850000 B_{0} B_{1} D_{7} + 1378561781250 B_{0} C_{4}^2 \\&\quad \quad + 153103668750 B_{0} C_{4} \lambda _{2}- 895916700000 B_{0} D_{7} \lambda _{1} +31396207500 B_{0} \lambda _{2}^2 \\&\quad \quad + 589866705000 B_{1}^2 D_{6} + 1469261531250 B_{1} C_{3} C_{4}-361894736250 B_{1} C_{3} \lambda _{2} \\&\quad \quad - 215667900 B_{1} C_{4} \lambda _{3} - 4489943805000 B_{1} D_{6} \lambda _{1}- 58609980 B_{1} \lambda _{2} \lambda _{3}\\&\quad \quad - 39322943437500 C_{2} D_{6} + 2015784984375 C_{3} C_{4} \lambda _{1} - 34459602187500 C_{3} D_{5} \\&\quad \quad + 638930986875 C_{3} \lambda _{1} \lambda _{2}- 8417557968750 C_{4} D_{4} +521805825 C_{4} \lambda _{1} \lambda _{3}\\&\quad \quad + 3352876031250 D_{4} \lambda _{2} + 7800300000 D_{5} \lambda _{3}-6183742500 D_{6} \lambda _{1}^2 \\&\quad \quad + 169387065 \lambda _{1} \lambda _{2} \lambda _{3},\\ \lambda _{14}&= - 607500 B_{0yy} + 2250 B_{0y} B_{1p} + 8910 B_{0y} A_{2} B_{1} + 7560 B_{0y} A_{2} \lambda _{1} + 66375 B_{0y} C_{3} \\&\quad \quad - 22 B_{0y} \lambda _{3} + 20250 B_{1x} B_{1y} + 9450 B_{1y} A_{2} B_{0} - 1350 B_{1y} B_{1}^2 - 90 B_{1y} B_{1} \lambda _{1} \\&\quad \quad + 50625 B_{1y} C_{2} - 855 B_{1y} \lambda _{1}^2,\\ \lambda _{15}&=37751298716250 A_{2x} A_{2}^2 - 62207526150000 A_{2x} C_{4} -16013792826000 A_{2x} \lambda _{2}\\&\quad \quad - 68496181668000 B_{1p} A_{2}^2 -104970815212500 B_{1p} C_{4} + 24473836354500 B_{1p} \lambda _{2}\\&\quad \quad - 892421479125000 C_{3pp} - 397004880825000 C_{3p} A_{2} - 477362777625000 C_{4px}\\&\quad \quad +22569136515000 C_{4p} B_{1} + 72626512005000 C_{4p} \lambda _{1} + 640641841132500 C_{4x} A_{2}\\&\quad \quad - 2182148140500000 C_{4y} - 3259278445500000 D_{5p} +627907383000000 D_{6x}\\&\quad \quad - 39656968964910 A_{2}^3 B_{1} + 19619486734770 A_{2}^3 \lambda _{1} - 204264974436750 A_{2}^2 C_{3}\\&\quad \quad - 80938338803 A_{2}^2 \lambda _{3} - 121019007663000 A_{2} B_{1} C_{4} + 285345314460 A_{2} B_{1} \lambda _{2}\\&\quad \quad + 119391630975750 A_{2} C_{4} \lambda _{1} -174315958267500 A_{2} D_{5} + 16210078823970 A_{2} \lambda _{1} \lambda _{2}\\&\quad \quad - 97674481800000 B_{0} D_{7}+ 45396652500000 B_{1} D_{6} - 613897474218750 C_{3} C_{4}\\&\quad \quad + 49679748960750 C_{3} \lambda _{2}-188416432400 C_{4} \lambda _{3} + 966289755150000 D_{6} \lambda _{1}\\&\quad \quad - 22084929912 \lambda _{2} \lambda _{3},\\ \end{aligned}$$

$$\begin{aligned} \lambda _{16}&=- 12456195187500 A_{2x} B_{1p} -2036878852500 A_{2x} A_{2} B_{1} - 3263539815000 A_{2x} A_{2} \lambda _{1}\\&\quad -26952929718750 A_{2x} C_{3} + 5036539500 A_{2x} \lambda _{3} - 2544574500000 B_{1px} A_{2}\\&\quad + 3193526250000 B_{1p}^2 + 3077757202500 B_{1p} A_{2} B_{1} -186084675000 B_{1p} A_{2} \lambda _{1}\\&\quad + 22532686312500 B_{1p} C_{3} + 4804800750 B_{1p} \lambda _{3} + 3124140435000 B_{1x} A_{2}^2\\&\quad - 2576559375000 B_{1x} C_{4} -895412475000 B_{1x} \lambda _{2} - 206612268750 B_{1y} A_{2}\\&\quad + 3412540125000 C_{3p} \lambda _{1} + 6738830437500 C_{3x} A_{2} - 50611451484375 C_{3y}\\&\quad + 2275026750000 C_{4p} B_{0} - 102376203750000 C_{4xx} + 17745208650000 C_{4x} B_{1}\\&\quad +227502675000 C_{4x} \lambda _{1} - 102376203750000 D_{5x} - 13753201612500 \lambda _{1y} A_{2}\\&\quad - 29108970000 \lambda _{3y} + 1932919443000 A_{2}^3 B_{0} + 107599889700 A_{2}^2 B_{1}^2\\&\quad + 470770566300 A_{2}^2 B_{1} \lambda _{1} + 10863840487500 A_{2}^2 C_{2} -313891459200 A_{2}^2 \lambda _{1}^2\\&\quad - 5119417687500 A_{2} B_{0} C_{4} - 383931495000 A_{2} B_{0} \lambda _{2} + 7971934241250 A_{2} B_{1} C_{3}\\&\quad + 970218270 A_{2} B_{1} \lambda _{3} + 1426857390000 A_{2} C_{3} \lambda _{1} + 39300800062500 A_{2} D_{4}\\&\quad + 802858470 A_{2} \lambda _{1} \lambda _{3} + 13650160500000 B_{0} D_{6} - 101232585000 B_{1}^2 C_{4}\\&\quad + 59694165000 B_{1}^2 \lambda _{2} + 147952980000 B_{1} C_{4} \lambda _{1} + 10920128400000 B_{1} D_{5}\\&\quad + 13079718000 B_{1} \lambda _{1} \lambda _{2} - 23504099062500 C_{2} C_{4} - 2238531187500 C_{2} \lambda _{2}\\&\quad + 34976323968750 C_{3}^2 + 13282365375 C_{3} \lambda _{3} +639627637500 C_{4} \lambda _{1}^2\\&\quad + 12740149800000 D_{5} \lambda _{1} + 25672828500 \lambda _{1}^2 \lambda _{2} +1101958 \lambda _{3}^2,\\ \lambda _{17}&=- 191362500 A_{2x} B_{1x} - 89302500 A_{2x} A_{2} B_{0} + 12757500 A_{2x} B_{1}^2 + 850500 A_{2x} B_{1} \lambda _{1}\\&\quad - 478406250 A_{2x} C_{2} + 8079750 A_{2x} \lambda _{1}^2 - 40095000 B_{0y} A_{2} + 28350000 B_{1p} B_{1x} \\&\quad + 13230000 B_{1p} A_{2} B_{0} -1890000 B_{1p} B_{1}^2 - 126000 B_{1p} B_{1} \lambda _{1} + 70875000 B_{1p} C_{2} \\&\quad -1197000 B_{1p} \lambda _{1}^2 + 273375000 B_{1xy} + 26365500 B_{1x} A_{2} B_{1} - 11056500 B_{1x} A_{2} \lambda _{1} \\&\quad + 134662500 B_{1x} C_{3} + 34650 B_{1x} \lambda _{3} -202500 B_{1y} B_{1} + 10732500 B_{1y} \lambda _{1} \\&\quad + 12303900 A_{2}^2 B_{0} B_{1} -5159700 A_{2}^2 B_{0} \lambda _{1} - 42 A_{2}^2 \lambda _{6} + 62842500 A_{2} B_{0} C_{3}\\&\quad + 16170 A_{2} B_{0} \lambda _{3} - 1757700 A_{2} B_{1}^3 + 619920 A_{2} B_{1}^2 \lambda _{1} + 65913750 A_{2} B_{1} C_{2} \\&\quad -1064070 A_{2} B_{1} \lambda _{1}^2- 27641250 A_{2} C_{2} \lambda _{1} + 466830 A_{2} \lambda _{1}^3 - 8977500 B_{1}^2 C_{3}\\&\quad - 2310 B_{1}^2 \lambda _{3} - 598500 B_{1} C_{3} \lambda _{1}- 154 B_{1} \lambda _{1} \lambda _{3} +336656250 C_{2} C_{3} + 86625 C_{2} \lambda _{3}\\&\quad - 5685750 C_{3} \lambda _{1}^2 - 350 C_{4} \lambda _{6} - 1463 \lambda _{1}^2 \lambda _{3} + 240 \lambda _{2} \lambda _{6} + 1923750 \lambda _{4},\\ \end{aligned}$$

$$\begin{aligned} \lambda _{18}&=2843100000 B_{0y} B_{1} - 601425000 B_{0y} \lambda _{1} + 2250 B_{1p} \lambda _{6} +2460375000 B_{1y} B_{0}\\&\quad - 55358437500 C_{1y} - 607500 \lambda _{6y} + 8910 A_{2} B_{1} \lambda _{6} + 7560 A_{2} \lambda _{1} \lambda _{6} + 66375 C_{3} \lambda _{6}\\&\quad - 22 \lambda _{3} \lambda _{6},\\ \lambda _{19}&=- 86495850000 A_{2xx} +21065184000 A_{2x} B_{1} - 2020302000 A_{2x} \lambda _{1} + 70567200000 B_{1px}\\&\quad + 2868786000 B_{1p} B_{1} - 4418511750 B_{1p} \lambda _{1} + 48844579500 B_{1x} A_{2} - 50243895000 B_{1y}\\&\quad + 185234343750 C_{3x} + 204719703750 \lambda _{1y} + 8306898300 A_{2}^2 B_{0} + 1899168444 A_{2} B_{1}^2\\&\quad - 4495733604 A_{2} B_{1} \lambda _{1} +28195836750 A_{2} C_{2} + 736957116 A_{2} \lambda _{1}^2 - 8120891250 B_{0} C_{4} \\&\quad - 1259793000 B_{0} \lambda _{2}+ 16156221750 B_{1} C_{3} + 1728164 B_{1} \lambda _{3} - 22174079625 C_{3} \lambda _{1} \\&\quad -176996643750 D_{4} - 6761957 \lambda _{1} \lambda _{3},\\ \lambda _{20}&=148837500 B_{0y} + 472500 B_{1p} B_{0} + 27337500 B_{1xx} - 6358500 B_{1x} B_{1} +3483000 B_{1x} \lambda _{1}\\&\quad - 1015200 A_{2} B_{0} B_{1} + 1479600 A_{2} B_{0} \lambda _{1} - 32805000 A_{2} C_{1} - 396 A_{2} \lambda _{6} - 29497500 B_{0} C_{3} \\&\quad + 1320 B_{0} \lambda _{3}+ 250884 B_{1}^3 -457956 B_{1}^2 \lambda _{1} - 19723500 B_{1} C_{2} + 382968 B_{1} \lambda _{1}^2 \\&\quad + 10773000 C_{2} \lambda _{1} +1558237500 D_{3}- 187668 \lambda _{1}^3 + 19 \lambda _{5},\\ \lambda _{21}&=-11400841210500 C_{4p} A_{2} + 188543797200000 D_{6p} + 258696086850000 D_{7x} \\&\quad - 3007393463946 A_{2}^4- 765008409625 A_{2}^2 C_{4} + 304249907613 A_{2}^2 \lambda _{2} \\&\quad - 23788045980000 A_{2} D_{6} - 9810740574000 B_{1} D_{7}+ 35034911057500 C_{4}^2 \\&\quad +5181010465800 C_{4} \lambda _{2} - 24354954288000 D_{7} \lambda _{1} + 465240734460 \lambda _{2}^2,\\ \lambda _{22}&=- 228549290762250 A_{2x} A_{2} +165925065678750 B_{1p} A_{2} + 30635038096875 C_{3p}\\&\quad - 1421989969725000 C_{4x} + 96089923759104 A_{2}^2 B_{1} - 45056084902452 A_{2}^2 \lambda _{1}\\&\quad +498027036101625 A_{2} C_{3}+ 195801538621 A_{2} \lambda _{3} + 86015883639375 B_{1} C_{4} \\&\quad -861988675500 B_{1} \lambda _{2} - 73447722556875 C_{4} \lambda _{1} - 1272332391046875 D_{5}\\&\quad -9202561540875 \lambda _{1} \lambda _{2},\\ \lambda _{23}&=911250 B_{1x} B_{0} - 20503125 C_{1x} + 425250 A_{2} B_{0}^2 - 60750 B_{0} B_{1}^2 + 20250 B_{0} B_{1} \lambda _{1}\\&\quad + 5467500 B_{0} C_{2} + 1366875 B_{1} C_{1} + 52 B_{1} \lambda _{6} - 455625 C_{1} \lambda _{1} - 205031250 D_{2} \\&\quad - 11 \lambda _{1} \lambda _{6},\\ \lambda _{24}&=145439214862500 C_{4p} + 36098747795172 A_{2}^3 - 92033790062000 A_{2} C_{4} \\&\quad - 13871260028325 A_{2} \lambda _{2} + 871341662728125 D_{6},\\ \lambda _{25}&=60750 A_{2x} - 9000 B_{1p} - 8370 A_{2} B_{1} + 3510 A_{2} \lambda _{1} - 42750 C_{3} - 11 \lambda _{3}. \end{aligned}$$