Sufficient Conditions for First-Order Differential Operators to be Associated with a q-Metamonogenic Operator in a Clifford Type Algebra

Abstract

Consider the initial value problem

$$\begin{aligned} \partial _{t}u= & {} {\mathcal L}(t,x,u,\partial _{x_{i}}u),\nonumber \\ u(0,x)= & {} \varphi (x), \end{aligned}$$

(0.1)

where t is the time, ${\mathcal L}$ is a linear first-order differential operator and $\varphi$ is a generalized q-metamonogenic function. This problem can be solved by applying the method of associated spaces which is constructed by Tutschke (see Solution of initial value problems in classes of generalized analytic functions, Teubner Leipzig and Springer, New York, 1989). In this work, we formulate sufficient conditions on the coefficients of the operator ${\mathcal L}$ under which this operator is associated to the space of generalized q-metamonogenic functions satisfying a differential equation with anti-q-metamonogenic right-hand side, when q and $\lambda$ are constant Clifford vectors. We also build a computational algorithm to check the computations in the cases ${\mathcal A}^{*}_{2,2}$ and ${\mathcal A}^{*}_{3,2}$. In conical domains, the initial value problem (0.1) is uniquely solvable for an operator ${\mathcal L}$ and for any generalized q-metamonogenic initial function $\varphi$, provided an interior estimate holds for generalized q-metamonogenic functions satisfying a differential equation with anti-q-metamonogenic right-hand side. The solution is also a generalized q-metamonogenic function for each fixed t. This work generalizes the results given in Di Teodoro and Sapian (Adv. Appl. Clifford Algebras, 25:283–301, 2015) and Van (Differential operator in a Clifford analysis associated to differential equations with anti-monogenic right hand side, IC/2006/134, 2016).

Appendices

Appendix 1

The code displayed below is to calculate the sufficient conditions for associated operators in the Clifford type Algebra ${\mathcal A}^{*}_{n,2}$ for $n=1,2,3$. The idea is to represent a function $u \in {\mathcal A}^{*}_{n,2}$ as a vector function $\mathbf {u} \in {\mathbf {R}}^m$, where $m=2^n-1$, with the same coordinates of u in the order prescribed in $\Gamma$ (see Remark 3). It is easy to see that the operation $e_i \cdot u$ for $i = 1,2,\ldots ,n$ can be represented as the matrix-vector operation $E_i\mathbf {u}$, where $E_i \in M_{m\times m}({\mathbf {R}})$. More details can be found in [5] for classic Clifford algebras. Using this fact, the differential operators l and ${\mathcal L}$ are written in this new format and $l ({\mathcal L}u)=0$ is computed directly using matrix algebra and the necessary conditions are found with the help of a Computer Algebra System.

Note that our algorithm:

• Naturally covers the cases ${\mathcal A}^{*}_{1,2}$ and ${\mathcal A}^{*}_{2,2}$ allowing us to verify the manual calculation.

• Works for any case ${\mathcal A}^{*}_{n,2}$ provided that the matrices $E_i$ for $i = 1,2,\ldots ,n$ are given. We are working on an algorithm to construct these matrices for all n.

Appendix 2

The coefficients $a_{ij}$ of Theorem 2 are given by

$$\begin{aligned} {{ a}_{00}}= & {} {{ s}_0}+k_{0} \\ {{ a}_{01}}= & {} {\alpha } _{1}\,{{ s}_1}+k_{1}\\ {{ a}_{02}}= & {} {\alpha }_{2}\,{{ s}_2}-2 \,\gamma _{12}\,{{ s}_1}+k_{2}\\ {{ a}_{03}}= & {} -{\alpha }_{3}\, {{ s}_3}+2\,\gamma _{23}\,{{ s}_2}+2\,\gamma _{13}\,{{ s}_1}+k_{3} \\ {{ a}_{04}}= & {} k_{4}-{\alpha }_{1}\,{\alpha }_{2}\, {{ s}_4}\\\\ {{ a}_{05}}= & {} -{\alpha }_{1}\,{\alpha }_{3}\, {{ s}_5}+2\,{\alpha }_{1}\,\gamma _{23}\,{{ s}_4}+k_{5}\\ {{ a}_{06}}= & {} -{\alpha }_{2}\,{\alpha }_{3}\,{{ s}_6}+2\, {\alpha }_{3}\,\gamma _{12}\,{{ s}_5}-4\,\gamma _{12}\,\gamma _{23 }\,{{ s}_4}-2\,{\alpha }_{2}\,\gamma _{13}\,{{ s}_4}+k_{6} \\ {{ a}_{07}}= & {} k_{7}-{\alpha }_{1}\,{\alpha }_{2}\, {\alpha }_{3}\,{{ s}_7} \\ {{ a}_{10}}= & {} {{ s}_1}-{{k_{1} }\over {{\alpha }_{1}}}\\ {{ a}_{11}}= & {} k_{0}-{{ s}_0}\\ {{ a}_{12}}= & {} -{\alpha }_{2}\,{{ s}_4}-{{k_{4}}\over { {\alpha }_{1}}}\\ {{ a}_{13}}= & {} {\alpha }_{3}\,{{ s}_5}-2\, \gamma _{23}\,{{ s}_4}-{{k_{5}}\over {{\alpha }_{1}}}\\ {{ a}_{14}}= & {} -{\alpha }_{2}\,{{ s}_2}+2\,\gamma _{12}\,{{ s}_1} +k_{2}\\ {{ a}_{15}}= & {} -{\alpha }_{3}\,{{ s}_3}+2\,\gamma _{23} \,{{ s}_2}+2\,\gamma _{13}\,{{ s}_1}+k_{3}\\ {{ a}_{16}}= & {} - {\alpha }_{2}\,{\alpha }_{3}\,{{ s}_7}-{{k_{7}}\over { {\alpha }_{1}}}\\ {{ a}_{17}}= & {} {\alpha }_{2}\,{\alpha } _{3}\,{{ s}_6}-2\,{\alpha }_{3}\,\gamma _{12}\,{{ s}_5}+4\, \gamma _{12}\,\gamma _{23}\,{{ s}_4}+2\,{\alpha }_{2}\,\gamma _{13 }\,{{ s}_4}+k_{6} \end{aligned}$$

$$\begin{aligned} {{ a}_{20}}= & {} {{ s}_2}-{{2\,k_{1}\,\gamma _{12} }\over {{\alpha }_{1}\,{\alpha }_{2}}}-{{k_{2}}\over { {\alpha }_{2}}}\\ {{ a}_{21}}= & {} {\alpha }_{1}\,{{ s}_4}+{{k _{4}}\over {{\alpha }_{2}}}\\ {{ a}_{22}}= & {} -2\,\gamma _{12}\, {{ s}_4}-{{ s}_0}-{{2\,k_{4}\,\gamma _{12}}\over {{\alpha }_{1} \,{\alpha }_{2}}}+k_{0}\\ {{ a}_{23}}= & {} {\alpha }_{3}\, {{ s}_6}+2\,\gamma _{13}\,{{ s}_4}-{{2\,k_{5}\,\gamma _{12}}\over { {\alpha }_{1}\,{\alpha }_{2}}}-{{k_{6}}\over {{\alpha }_{ 2}}}\\ {{ a}_{24}}= & {} {\alpha }_{1}\,{{ s}_1}-k_{1}\\ {{ a}_{25}}= & {} {\alpha }_{1}\,{\alpha }_{3}\,{{ s}_7}+{{k_{7} }\over {{\alpha }_{2}}}\\ {{ a}_{26}}= & {} -2\,{\alpha }_{3}\, \gamma _{12}\,{{ s}_7}-{\alpha }_{3}\,{{ s}_3}+2\,\gamma _{23}\, {{ s}_2}+2\,\gamma _{13}\,{{ s}_1}-{{2\,k_{7}\,\gamma _{12}}\over { {\alpha }_{1}\,{\alpha }_{2}}}+k_{3}\\ {{ a}_{27}}= & {} - {\alpha }_{1}\,{\alpha }_{3}\,{{ s}_5}+2\,{\alpha }_{1} \,\gamma _{23}\,{{ s}_4}-k_{5}\\ {{ a}_{30}}= & {} {{ s}_3}-{{4\,k_{1} \,\gamma _{12}\,\gamma _{23}}\over {{\alpha }_{1}\,{\alpha }_{2 }\,{\alpha }_{3}}}-{{2\,k_{2}\,\gamma _{23}}\over {{\alpha }_{ 2}\,{\alpha }_{3}}}-{{2\,k_{1}\,\gamma _{13}}\over {{\alpha } _{1}\,{\alpha }_{3}}}-{{k_{3}}\over {{\alpha }_{3}}}\\ {{ a}_{31}}= & {} {\alpha }_{1}\,{{ s}_5}+{{2\,k_{4}\,\gamma _{23} }\over {{\alpha }_{2}\,{\alpha }_{3}}}+{{k_{5}}\over { {\alpha }_{3}}}\\ {{ a}_{32}}= & {} {\alpha }_{2}\,{{ s}_6}-2\, \gamma _{12}\,{{ s}_5}-{{4\,k_{4}\,\gamma _{12}\,\gamma _{23}}\over { {\alpha }_{1}\,{\alpha }_{2}\,{\alpha }_{3}}}-{{2\,k_{4} \,\gamma _{13}}\over {{\alpha }_{1}\,{\alpha }_{3}}}+{{k_{6} }\over {{\alpha }_{3}}}\\ {{ a}_{33}}= & {} 2\,\gamma _{23}\,{{ s}_6} +{{ s}_0}+{{4\,k_{4}\,\gamma _{13}\,\gamma _{23}}\over {{\alpha } _{1}\,{\alpha }_{2}\,{\alpha }_{3}}}-{{4\,k_{5}\,\gamma _{12} \,\gamma _{23}}\over {{\alpha }_{1}\,{\alpha }_{2}\, {\alpha }_{3}}}-{{2\,k_{6}\,\gamma _{23}}\over {{\alpha }_{2} \,{\alpha }_{3}}}+k_{0}\\ {{ a}_{34}}= & {} -{\alpha }_{1}\, {\alpha }_{2}\,{{ s}_7}-{{k_{7}}\over {{\alpha }_{3}}}\\ {{ a}_{35}}= & {} 2\,{\alpha }_{1}\,\gamma _{23}\,{{ s}_7}-2\, {\alpha }_{1}\,\gamma _{13}\,{{ s}_5}+{\alpha }_{1}\, {{ s}_1}+{{4\,k_{4}\,\gamma _{13}\,\gamma _{23}}\over {{\alpha }_{ 2}\,{\alpha }_{3}}}+{{2\,k_{7}\,\gamma _{23}}\over {{\alpha } _{2}\,{\alpha }_{3}}}+{{2\,k_{5}\,\gamma _{13}}\over { {\alpha }_{3}}}-k_{1} \end{aligned}$$

$$\begin{aligned} {{ a}_{36}}\;= & {} \; - 4{} {\gamma _{12}}{} {\gamma _{23}}{} {{ s}_7} - 2{} {\alpha _2}{} {\gamma _{13}}{} {{ s}_7} + 4{} {\gamma _{12}}{} {\gamma _{13}}{} {{ s}_5} + {\alpha _2}{} {{ s}_2} - 2{} {\gamma _{12}}{} {{ s}_1}\mathrm{{ }} \\&-\, \frac{{8{} {k_4}{} {\gamma _{12}}{} {\gamma _{13}}{} {\gamma _{23}}}}{{{\alpha _1}{} {\alpha _2}{} {\alpha _3}}} - \frac{{4{} {k_7}{} {\gamma _{12}}{} {\gamma _{23}}}}{{{\alpha _1}{} {\alpha _2}{} {\alpha _3}}} - \frac{{4{} {k_5}{} {\gamma _{12}}{} {\gamma _{13}}}}{{{\alpha _1}{} {\alpha _3}}} - \frac{{2{} {k_7}{} {\gamma _{13}}}}{{{\alpha _1}{} {\alpha _3}}} - {k_2} \\ {{ a}_{37}}= & {} \;\;{\alpha _1}{} {\alpha _2}{} {{ s}_4} + {k_4} \\ {{ a}_{40}}= & {} {{ s}_4}-{{k_{4}}\over {{\alpha }_{1}\, {\alpha }_{2}}}\\ {{ a}_{41}}= & {} -{{ s}_2}-{{2\,k_{1}\,\gamma _{ 12}}\over {{\alpha }_{1}\,{\alpha }_{2}}}-{{k_{2}}\over { {\alpha }_{2}}}\\ {{ a}_{42}}= & {} {{ s}_1}+{{k_{1}}\over { {\alpha }_{1}}}\\ {{ a}_{43}}= & {} -{\alpha }_{3}\,{{ s}_7}-{{ k_{7}}\over {{\alpha }_{1}\,{\alpha }_{2}}}\\ {{ a}_{44}}= & {} 2 \,\gamma _{12}\,{{ s}_4}+{{ s}_0}-{{2\,k_{4}\,\gamma _{12}}\over { {\alpha }_{1}\,{\alpha }_{2}}}+k_{0}\\ \end{aligned}$$

$$\begin{aligned} {{ a}_{45}}= & {} {\alpha }_{3}\,{{ s}_6}+2\,\gamma _{13}\,{{ s}_4}-{{2\,k_{5}\, \gamma _{12}}\over {{\alpha }_{1}\,{\alpha }_{2}}}-{{k_{6} }\over {{\alpha }_{2}}}\\ {{ a}_{46}}= & {} -{\alpha }_{3}\, {{ s}_5}+2\,\gamma _{23}\,{{ s}_4}+{{k_{5}}\over {{\alpha }_{1} }}\\ {{ a}_{47}}= & {} 2\,{\alpha }_{3}\,\gamma _{12}\,{{ s}_7}+ {\alpha }_{3}\,{{ s}_3}-2\,\gamma _{23}\,{{ s}_2}-2\,\gamma _{13 }\,{{ s}_1}-{{2\,k_{7}\,\gamma _{12}}\over {{\alpha }_{1}\, {\alpha }_{2}}}+k_{3}\\ {{ a}_{50}}= & {} {{ s}_5}-{{2\,k_{4}\, \gamma _{23}}\over {{\alpha }_{1}\,{\alpha }_{2}\, {\alpha }_{3}}}-{{k_{5}}\over {{\alpha }_{1}\,{\alpha }_{ 3}}}\\ {{ a}_{51}}= & {} -{{ s}_3}-{{4\,k_{1}\,\gamma _{12}\,\gamma _{23} }\over {{\alpha }_{1}\,{\alpha }_{2}\,{\alpha }_{3}}}-{{2 \,k_{2}\,\gamma _{23}}\over {{\alpha }_{2}\,{\alpha }_{3}}}-{{ 2\,k_{1}\,\gamma _{13}}\over {{\alpha }_{1}\,{\alpha }_{3}}}- {{k_{3}}\over {{\alpha }_{3}}}\\ {{ a}_{52}}= & {} {{k_{7}}\over { {\alpha }_{1}\,{\alpha }_{3}}}-{\alpha }_{2}\,{{ s}_7} \\ {{ a}_{53}}= & {} -2\,\gamma _{23}\,{{ s}_7}+2\,\gamma _{13}\, {{ s}_5}-{{ s}_1}-{{4\,k_{4}\,\gamma _{13}\,\gamma _{23}}\over { {\alpha }_{1}\,{\alpha }_{2}\,{\alpha }_{3}}}-{{2\,k_{7} \,\gamma _{23}}\over {{\alpha }_{1}\,{\alpha }_{2}\, {\alpha }_{3}}}-{{2\,k_{5}\,\gamma _{13}}\over {{\alpha }_{1} \,{\alpha }_{3}}}+{{k_{1}}\over {{\alpha }_{1}}} \end{aligned}$$

$$\begin{aligned} {{ a}_{54}}= & {} -{\alpha }_{2}\,{{ s}_6}+2\,\gamma _{12}\,{{ s}_5} -{{4\,k_{4}\,\gamma _{12}\,\gamma _{23}}\over {{\alpha }_{1}\, {\alpha }_{2}\,{\alpha }_{3}}}-{{2\,k_{4}\,\gamma _{13} }\over {{\alpha }_{1}\,{\alpha }_{3}}}+{{k_{6}}\over { {\alpha }_{3}}}\\ {{ a}_{55}}= & {} 2\,\gamma _{23}\,{{ s}_6}+ {{ s}_0}+{{4\,k_{4}\,\gamma _{13}\,\gamma _{23}}\over {{\alpha }_{ 1}\,{\alpha }_{2}\,{\alpha }_{3}}}-{{4\,k_{5}\,\gamma _{12}\, \gamma _{23}}\over {{\alpha }_{1}\,{\alpha }_{2}\, {\alpha }_{3}}}-{{2\,k_{6}\,\gamma _{23}}\over {{\alpha }_{2} \,{\alpha }_{3}}}+k_{0} \\ {{ a}_{56}}= & {} {\alpha }_{2}\, {{ s}_4}-{{k_{4}}\over {{\alpha }_{1}}} \\ {{ a}_{57}}= & {} 4\, \gamma _{12}\,\gamma _{23}\,{{ s}_7}+2\,{\alpha }_{2}\,\gamma _{13 }\,{{ s}_7}-4\,\gamma _{12}\,\gamma _{13}\,{{ s}_5}-{\alpha }_{2 }\,{{ s}_2}+2\,\gamma _{12}\,{{ s}_1} \\&-\, {{8\,k_{4}\,\gamma _{12}\, \gamma _{13}\,\gamma _{23}}\over {{\alpha }_{1}\,{\alpha }_{2} \,{\alpha }_{3}}}-{{4\,k_{7}\,\gamma _{12}\,\gamma _{23}}\over { {\alpha }_{1}\,{\alpha }_{2}\,{\alpha }_{3}}}-{{4\,k_{5} \,\gamma _{12}\,\gamma _{13}}\over {{\alpha }_{1}\,{\alpha }_{3 }}}-{{2\,k_{7}\,\gamma _{13}}\over {{\alpha }_{1}\,{\alpha }_{ 3}}}-k_{2} \end{aligned}$$

$$\begin{aligned} {{ a}_{60}}= & {} {{ s}_6}+{{2\,k_{4}\,\gamma _{13}}\over { {\alpha }_{1}\,{\alpha }_{2}\,{\alpha }_{3}}}-{{2\,k_{5} \,\gamma _{12}}\over {{\alpha }_{1}\,{\alpha }_{2}\, {\alpha }_{3}}}-{{k_{6}}\over {{\alpha }_{2}\,{\alpha }_{ 3}}}\\ {{ a}_{61}}= & {} {\alpha }_{1}\,{{ s}_7}-{{k_{7}}\over { {\alpha }_{2}\,{\alpha }_{3}}}\\ {{ a}_{62}}= & {} -2\,\gamma _{ 12}\,{{ s}_7}-{{ s}_3}-{{4\,k_{1}\,\gamma _{12}\,\gamma _{23}}\over { {\alpha }_{1}\,{\alpha }_{2}\,{\alpha }_{3}}}-{{2\,k_{2} \,\gamma _{23}}\over {{\alpha }_{2}\,{\alpha }_{3}}}-{{2\,k_{1 }\,\gamma _{13}}\over {{\alpha }_{1}\,{\alpha }_{3}}}+{{2\,k_{ 7}\,\gamma _{12}}\over {{\alpha }_{1}\,{\alpha }_{2}\, {\alpha }_{3}}}-{{k_{3}}\over {{\alpha }_{3}}}\\ {{ a}_{63}}= & {} 2\,\gamma _{13}\,{{ s}_7}-{{ s}_2}+{{2\,k_{7}\,\gamma _{13}}\over {{\alpha }_{1}\,{\alpha }_{2}\,{\alpha }_{3} }}+{{2\,k_{1}\,\gamma _{12}}\over {{\alpha }_{1}\,{\alpha }_{2 }}}+{{k_{2}}\over {{\alpha }_{2}}}\\ {{ a}_{64}}= & {} {\alpha } _{1}\,{{ s}_5}-{{2\,k_{4}\,\gamma _{23}}\over {{\alpha }_{2}\, {\alpha }_{3}}}-{{k_{5}}\over {{\alpha }_{3}}}\\ {{ a}_{65}}= & {} {{k_{4}}\over {{\alpha }_{2}}}-{\alpha }_{1}\, {{ s}_4}\\ {{ a}_{66}}= & {} 2\,\gamma _{23}\,{{ s}_6}{+}2\,\gamma _{12}\, {{ s}_4}{+}{{ s}_0}{+}{{4\,k_{4}\,\gamma _{13}\,\gamma _{23}}\over { {\alpha }_{1}\,{\alpha }_{2}\,{\alpha }_{3}}}-{{4\,k_{5} \,\gamma _{12}\,\gamma _{23}}\over {{\alpha }_{1}\,{\alpha }_{2 }\,{\alpha }_{3}}}-{{2\,k_{6}\,\gamma _{23}}\over {{\alpha }_{ 2}\,{\alpha }_{3}}}-{{2\,k_{4}\,\gamma _{12}}\over {{\alpha } _{1}\,{\alpha }_{2}}}+k_{0}\\ {{ a}_{67}}= & {} 2\,{\alpha }_{1} \,\gamma _{23}\,{{ s}_7}-2\,{\alpha }_{1}\,\gamma _{13}\, {{ s}_5}+{\alpha }_{1}\,{{ s}_1}-{{4\,k_{4}\,\gamma _{13}\, \gamma _{23}}\over {{\alpha }_{2}\,{\alpha }_{3}}}-{{2\,k_{7} \,\gamma _{23}}\over {{\alpha }_{2}\,{\alpha }_{3}}}-{{2\,k_{5 }\,\gamma _{13}}\over {{\alpha }_{3}}}+k_{1} \end{aligned}$$

$$\begin{aligned} {{ a}_{70}}= & {} {{ s}_7}+{{k_{7}}\over {{\alpha }_{1}\,{\alpha }_{2}\, {\alpha }_{3}}}\\ {{ a}_{71}}= & {} -{{ s}_6}+{{2\,k_{4}\,\gamma _{ 13}}\over {{\alpha }_{1}\,{\alpha }_{2}\,{\alpha }_{3}}}- {{2\,k_{5}\,\gamma _{12}}\over {{\alpha }_{1}\,{\alpha }_{2}\, {\alpha }_{3}}}-{{k_{6}}\over {{\alpha }_{2}\,{\alpha }_{ 3}}}\\ {{ a}_{72}}= & {} {{ s}_5}+{{2\,k_{4}\,\gamma _{23}}\over { {\alpha }_{1}\,{\alpha }_{2}\,{\alpha }_{3}}}+{{k_{5} }\over {{\alpha }_{1}\,{\alpha }_{3}}}\\ {{ a}_{73}}= & {} {{ s}_4}-{{k_{4}}\over {{\alpha }_{1}\,{\alpha }_{2}}}\\ {{ a}_{74}}= & {} 2\,\gamma _{12}\,{{ s}_7}+{{ s}_3}-{{4\,k_{1}\,\gamma _{12}\,\gamma _{23}}\over {{\alpha }_{1}\,{\alpha }_{2}\, {\alpha }_{3}}}-{{2\,k_{2}\,\gamma _{23}}\over {{\alpha }_{2} \,{\alpha }_{3}}}-{{2\,k_{1}\,\gamma _{13}}\over {{\alpha }_{1 }\,{\alpha }_{3}}}+{{2\,k_{7}\,\gamma _{12}}\over {{\alpha }_{ 1}\,{\alpha }_{2}\,{\alpha }_{3}}}-{{k_{3}}\over { {\alpha }_{3}}}\\ {{ a}_{75}}= & {} 2\,\gamma _{13}\,{{ s}_7}- {{ s}_2}+{{2\,k_{7}\,\gamma _{13}}\over {{\alpha }_{1}\, {\alpha }_{2}\,{\alpha }_{3}}}+{{2\,k_{1}\,\gamma _{12} }\over {{\alpha }_{1}\,{\alpha }_{2}}}+{{k_{2}}\over { {\alpha }_{2}}}\\ {{ a}_{76}}= & {} 2\,\gamma _{23}\,{{ s}_7}-2\, \gamma _{13}\,{{ s}_5}+{{ s}_1}+{{4\,k_{4}\,\gamma _{13}\,\gamma _{23 }}\over {{\alpha }_{1}\,{\alpha }_{2}\,{\alpha }_{3}}}+{{ 2\,k_{7}\,\gamma _{23}}\over {{\alpha }_{1}\,{\alpha }_{2}\, {\alpha }_{3}}}+{{2\,k_{5}\,\gamma _{13}}\over {{\alpha }_{1} \,{\alpha }_{3}}}-{{k_{1}}\over {{\alpha }_{1}}}\\ {{ a}_{77}}= & {} -2\,\gamma _{23}\,{{ s}_6}-2\,\gamma _{12}\,{{ s}_4}- {{ s}_0}{+}{{4\,k_{4}\,\gamma _{13}\,\gamma _{23}}\over {{\alpha }_{ 1}\,{\alpha }_{2}\,{\alpha }_{3}}}{-}{{4\,k_{5}\,\gamma _{12}\, \gamma _{23}}\over {{\alpha }_{1}\,{\alpha }_{2}\, {\alpha }_{3}}}{-}{{2\,k_{6}\,\gamma _{23}}\over {{\alpha }_{2} \,{\alpha }_{3}}}{-}{{2\,k_{4}\,\gamma _{12}}\over {{\alpha }_{1 }\,{\alpha }_{2}}}+k_{0} \end{aligned}$$

C is a matrix $\mathbf {R}^{8\times 8}$ with entries given by

$$\begin{aligned} {{ C}_{00}}= & {} 0\\ {{ C}_{01}}= & {} {{4\,k_{1}\,\gamma _{12}\, \gamma _{13}\,\gamma _{23}}\over {{\alpha }_{1}\,{\alpha }_{2} \,{\alpha }_{3}}}+{{2\,k_{2}\,\gamma _{13}\,\gamma _{23}}\over { {\alpha }_{2}\,{\alpha }_{3}}}+{{2\,k_{1}\,\gamma _{13}^2 }\over {{\alpha }_{1}\,{\alpha }_{3}}}+{{k_{3}\,\gamma _{13} }\over {{\alpha }_{3}}}-{{2\,k_{1}\,\gamma _{12}^2}\over { {\alpha }_{1}\,{\alpha }_{2}}}-{{k_{2}\,\gamma _{12}}\over { {\alpha }_{2}}}+k_{1}\\ {{ C}_{02}}= & {} {{4\,k_{1}\,\gamma _{12}\, \gamma _{23}^2}\over {{\alpha }_{1}\,{\alpha }_{2}\, {\alpha }_{3}}}+{{2\,k_{2}\,\gamma _{23}^2}\over {{\alpha }_{2 }\,{\alpha }_{3}}}+{{2\,k_{1}\,\gamma _{13}\,\gamma _{23}}\over { {\alpha }_{1}\,{\alpha }_{3}}}+{{k_{3}\,\gamma _{23}}\over { {\alpha }_{3}}}+{{k_{1}\,\gamma _{12}}\over {{\alpha }_{1}}}+k _{2}\\ {{ C}_{03}}= & {} -{{2\,k_{1}\,\gamma _{12}\,\gamma _{23}}\over { {\alpha }_{1}\,{\alpha }_{2}}}-{{k_{2}\,\gamma _{23}}\over { {\alpha }_{2}}}-{{k_{1}\,\gamma _{13}}\over {{\alpha }_{1}}} \\ {{ C}_{04}}= & {} {{2\,k_{4}\,\gamma _{23}^2}\over {{\alpha }_{2}\, {\alpha }_{3}}}+{{4\,k_{4}\,\gamma _{12}\,\gamma _{13}\,\gamma _{23 }}\over {{\alpha }_{1}\,{\alpha }_{2}\,{\alpha }_{3}}}-{{ 4\,k_{5}\,\gamma _{12}^2\,\gamma _{23}}\over {{\alpha }_{1}\, {\alpha }_{2}\,{\alpha }_{3}}}-{{2\,k_{6}\,\gamma _{12}\, \gamma _{23}}\over {{\alpha }_{2}\,{\alpha }_{3}}}+{{k_{5}\, \gamma _{23}}\over {{\alpha }_{3}}}\\&+\, {{2\,k_{4}\,\gamma _{13}^2 }\over {{\alpha }_{1}\,{\alpha }_{3}}}-{{2\,k_{5}\,\gamma _{12 }\,\gamma _{13}}\over {{\alpha }_{1}\,{\alpha }_{3}}}-{{k_{6} \,\gamma _{13}}\over {{\alpha }_{3}}}\\ {{ C}_{05}}= & {} -{{k_{4}\, \gamma _{23}}\over {{\alpha }_{2}}}-{{2\,k_{4}\,\gamma _{12}\, \gamma _{13}}\over {{\alpha }_{1}\,{\alpha }_{2}}}+{{2\,k_{5} \,\gamma _{12}^2}\over {{\alpha }_{1}\,{\alpha }_{2}}}+{{k_{6} \,\gamma _{12}}\over {{\alpha }_{2}}}\\ {{ C}_{06}}= & {} {{k_{4}\, \gamma _{13}}\over {{\alpha }_{1}}}-{{k_{5}\,\gamma _{12}}\over { {\alpha }_{1}}}\\ {{ C}_{07}}= & {} k_{7} \\ {{ C}_{10}}= & {} -{{k_{1} }\over {{\alpha }_{1}}}\\ {{ C}_{11}}= & {} {{2\,k_{4}\,\gamma _{13}\, \gamma _{23}}\over {{\alpha }_{1}\,{\alpha }_{2}\, {\alpha }_{3}}}+{{k_{5}\,\gamma _{13}}\over {{\alpha }_{1}\, {\alpha }_{3}}}+{{k_{4}\,\gamma _{12}}\over {{\alpha }_{1}\, {\alpha }_{2}}}\\ {{ C}_{12}}= & {} {{2\,k_{4}\,\gamma _{23}^2}\over { {\alpha }_{1}\,{\alpha }_{2}\,{\alpha }_{3}}}+{{k_{5}\, \gamma _{23}}\over {{\alpha }_{1}\,{\alpha }_{3}}}-{{k_{4} }\over {{\alpha }_{1}}}\\ {{ C}_{13}}= & {} -{{k_{4}\,\gamma _{23} }\over {{\alpha }_{1}\,{\alpha }_{2}}}-{{k_{5}}\over { {\alpha }_{1}}}\\ {{ C}_{14}}= & {} -{{4\,k_{1}\,\gamma _{12}\,\gamma _{23}^2}\over {{\alpha }_{1}\,{\alpha }_{2}\,{\alpha }_{3 }}}{-}{{2\,k_{2}\,\gamma _{23}^2}\over {{\alpha }_{2}\,{\alpha } _{3}}}{-}{{2\,k_{1}\,\gamma _{13}\,\gamma _{23}}\over {{\alpha }_{1} \,{\alpha }_{3}}}-{{2\,k_{7}\,\gamma _{12}\,\gamma _{23}}\over { {\alpha }_{1}\,{\alpha }_{2}\,{\alpha }_{3}}}{-}{{k_{3}\, \gamma _{23}}\over {{\alpha }_{3}}}\\&-{{k_{7}\,\gamma _{13}}\over { {\alpha }_{1}\,{\alpha }_{3}}}-{{k_{1}\,\gamma _{12}}\over { {\alpha }_{1}}}\\ \end{aligned}$$

$$\begin{aligned} {{ C}_{15}}= & {} {{2\,k_{1}\,\gamma _{12}\,\gamma _{23}}\over {{\alpha }_{1}\,{\alpha }_{2}}}+{{k_{2}\,\gamma _{ 23}}\over {{\alpha }_{2}}}+{{k_{1}\,\gamma _{13}}\over { {\alpha }_{1}}}+{{k_{7}\,\gamma _{12}}\over {{\alpha }_{1}\, {\alpha }_{2}}}+k_{3}\\ {{ C}_{16}}= & {} -{{k_{7}}\over { {\alpha }_{1}}}\\ {{ C}_{17}}= & {} {{k_{4}\,\gamma _{13}}\over { {\alpha }_{1}}}-{{k_{5}\,\gamma _{12}}\over {{\alpha }_{1}}}\\ {{ C}_{20}}= & {} -{{2\,k_{1}\,\gamma _{12}}\over {{\alpha }_{1}\, {\alpha }_{2}}}-{{k_{2}}\over {{\alpha }_{2}}}\\ {{ C}_{21}}= & {} -{{2\,k_{4}\,\gamma _{13}^2}\over {{\alpha }_{1}\, {\alpha }_{2}\,{\alpha }_{3}}}+{{2\,k_{5}\,\gamma _{12}\, \gamma _{13}}\over {{\alpha }_{1}\,{\alpha }_{2}\, {\alpha }_{3}}}+{{k_{6}\,\gamma _{13}}\over {{\alpha }_{2}\, {\alpha }_{3}}}+{{k_{4}}\over {{\alpha }_{2}}}\\ {{ C}_{22}}= & {} -{{2\,k_{4}\,\gamma _{13}\,\gamma _{23}}\over { {\alpha }_{1}\,{\alpha }_{2}\,{\alpha }_{3}}}+{{2\,k_{5} \,\gamma _{12}\,\gamma _{23}}\over {{\alpha }_{1}\,{\alpha }_{2 }\,{\alpha }_{3}}}+{{k_{6}\,\gamma _{23}}\over {{\alpha }_{2} \,{\alpha }_{3}}}-{{k_{4}\,\gamma _{12}}\over {{\alpha }_{1}\, {\alpha }_{2}}}\\ {{ C}_{23}}= & {} {{k_{4}\,\gamma _{13}}\over { {\alpha }_{1}\,{\alpha }_{2}}}-{{2\,k_{5}\,\gamma _{12} }\over {{\alpha }_{1}\,{\alpha }_{2}}}-{{k_{6}}\over { {\alpha }_{2}}}\\ {{ C}_{24}}= & {} {{4\,k_{1}\,\gamma _{12}\,\gamma _{13}\,\gamma _{23}}\over {{\alpha }_{1}\,{\alpha }_{2}\, {\alpha }_{3}}}+{{2\,k_{2}\,\gamma _{13}\,\gamma _{23}}\over { {\alpha }_{2}\,{\alpha }_{3}}}-{{k_{7}\,\gamma _{23}}\over { {\alpha }_{2}\,{\alpha }_{3}}}+{{2\,k_{1}\,\gamma _{13}^2 }\over {{\alpha }_{1}\,{\alpha }_{3}}}+{{k_{3}\,\gamma _{13} }\over {{\alpha }_{3}}}-{{2\,k_{1}\,\gamma _{12}^2}\over { {\alpha }_{1}\,{\alpha }_{2}}}\\&\quad -{{k_{2}\,\gamma _{12}}\over { {\alpha }_{2}}} \end{aligned}$$

$$\begin{aligned} {{ C}_{25}}= & {} {{k_{7}}\over {{\alpha }_{2} }}\\ {{ C}_{26}}= & {} {{2\,k_{1}\,\gamma _{12}\,\gamma _{23}}\over { {\alpha }_{1}\,{\alpha }_{2}}}+{{k_{2}\,\gamma _{23}}\over { {\alpha }_{2}}}+{{k_{1}\,\gamma _{13}}\over {{\alpha }_{1}}}- {{k_{7}\,\gamma _{12}}\over {{\alpha }_{1}\,{\alpha }_{2}}}+k _{3}\\ {{ C}_{27}}= & {} {{k_{4}\,\gamma _{23}}\over {{\alpha }_{2}}}+ {{2\,k_{4}\,\gamma _{12}\,\gamma _{13}}\over {{\alpha }_{1}\, {\alpha }_{2}}}-{{2\,k_{5}\,\gamma _{12}^2}\over {{\alpha }_{1 }\,{\alpha }_{2}}}-{{k_{6}\,\gamma _{12}}\over {{\alpha }_{2} }} \\ {{ C}_{30}}= & {} 0\\ {{ C}_{31}}= & {} {{2\,k_{4}\,\gamma _{23}}\over { {\alpha }_{2}\,{\alpha }_{3}}}+{{2\,k_{4}\,\gamma _{12}\, \gamma _{13}}\over {{\alpha }_{1}\,{\alpha }_{2}\, {\alpha }_{3}}}-{{2\,k_{5}\,\gamma _{12}^2}\over {{\alpha }_{1 }\,{\alpha }_{2}\,{\alpha }_{3}}}-{{k_{6}\,\gamma _{12} }\over {{\alpha }_{2}\,{\alpha }_{3}}}+{{k_{5}}\over { {\alpha }_{3}}}\\ {{ C}_{32}}= & {} -{{2\,k_{4}\,\gamma _{12}\,\gamma _{23}}\over {{\alpha }_{1}\,{\alpha }_{2}\,{\alpha }_{3} }}-{{2\,k_{4}\,\gamma _{13}}\over {{\alpha }_{1}\,{\alpha }_{3 }}}+{{k_{5}\,\gamma _{12}}\over {{\alpha }_{1}\,{\alpha }_{3} }}+{{k_{6}}\over {{\alpha }_{3}}}\\ {{ C}_{33}}= & {} {{2\,k_{4}\, \gamma _{13}\,\gamma _{23}}\over {{\alpha }_{1}\,{\alpha }_{2} \,{\alpha }_{3}}}-{{2\,k_{5}\,\gamma _{12}\,\gamma _{23}}\over { {\alpha }_{1}\,{\alpha }_{2}\,{\alpha }_{3}}}-{{k_{6}\, \gamma _{23}}\over {{\alpha }_{2}\,{\alpha }_{3}}}\\ {{ C}_{34}}= & {} -{{k_{7}}\over {{\alpha }_{3}}}\\ {{ C}_{35}}= & {} {{k _{7}\,\gamma _{23}}\over {{\alpha }_{2}\,{\alpha }_{3}}}-{{4\, k_{4}\,\gamma _{12}\,\gamma _{13}^2}\over {{\alpha }_{1}\, {\alpha }_{2}\,{\alpha }_{3}}}+{{4\,k_{5}\,\gamma _{12}^2\, \gamma _{13}}\over {{\alpha }_{1}\,{\alpha }_{2}\, {\alpha }_{3}}}+{{2\,k_{6}\,\gamma _{12}\,\gamma _{13}}\over { {\alpha }_{2}\,{\alpha }_{3}}}-{{2\,k_{1}\,\gamma _{12}^2 }\over {{\alpha }_{1}\,{\alpha }_{2}}}-{{k_{2}\,\gamma _{12} }\over {{\alpha }_{2}}}\\ {{ C}_{36}}= & {} {{4\,k_{1}\,\gamma _{12}\, \gamma _{23}^2}\over {{\alpha }_{1}\,{\alpha }_{2}\, {\alpha }_{3}}}+{{2\,k_{2}\,\gamma _{23}^2}\over {{\alpha }_{2 }\,{\alpha }_{3}}}-{{4\,k_{4}\,\gamma _{12}\,\gamma _{13}\,\gamma _{23}}\over {{\alpha }_{1}\,{\alpha }_{2}\,{\alpha }_{3} }}+{{2\,k_{1}\,\gamma _{13}\,\gamma _{23}}\over {{\alpha }_{1}\, {\alpha }_{3}}}-{{2\,k_{7}\,\gamma _{12}\,\gamma _{23}}\over { {\alpha }_{1}\,{\alpha }_{2}\,{\alpha }_{3}}}\\&+\,{{k_{3}\, \gamma _{23}}\over {{\alpha }_{3}}}-{{2\,k_{5}\,\gamma _{12}\, \gamma _{13}}\over {{\alpha }_{1}\,{\alpha }_{3}}}-{{k_{7}\, \gamma _{13}}\over {{\alpha }_{1}\,{\alpha }_{3}}}+{{k_{1}\, \gamma _{12}}\over {{\alpha }_{1}}}\\ {{ C}_{37}}= & {} {{2\,k_{4}\,\gamma _{23}^2}\over {{\alpha }_{2}\,{\alpha }_{3}}}+{{4\,k_{4 }\,\gamma _{12}\,\gamma _{13}\,\gamma _{23}}\over {{\alpha }_{1}\, {\alpha }_{2}\,{\alpha }_{3}}}-{{4\,k_{5}\,\gamma _{12}^2\, \gamma _{23}}\over {{\alpha }_{1}\,{\alpha }_{2}\, {\alpha }_{3}}}-{{2\,k_{6}\,\gamma _{12}\,\gamma _{23}}\over { {\alpha }_{2}\,{\alpha }_{3}}}+{{k_{5}\,\gamma _{23}}\over { {\alpha }_{3}}}\\&+\, {{2\,k_{4}\,\gamma _{13}^2}\over {{\alpha }_{1 }\,{\alpha }_{3}}}-{{2\,k_{5}\,\gamma _{12}\,\gamma _{13}}\over { {\alpha }_{1}\,{\alpha }_{3}}}-{{k_{6}\,\gamma _{13}}\over { {\alpha }_{3}}} \end{aligned}$$

$$\begin{aligned} {{ C}_{40}}= & {} 0\\ {{ C}_{41}}= & {} {{k_{7}\, \gamma _{13}}\over {{\alpha }_{1}\,{\alpha }_{2}\, {\alpha }_{3}}}\\ {{ C}_{42}}= & {} {{k_{7}\,\gamma _{23}}\over { {\alpha }_{1}\,{\alpha }_{2}\,{\alpha }_{3}}}\\ {{ C}_{43}}= & {} -{{k_{7}}\over {{\alpha }_{1}\,{\alpha }_{2}}} \\ {{ C}_{44}}= & {} {{2\,k_{5}\,\gamma _{12}\,\gamma _{23}}\over { {\alpha }_{1}\,{\alpha }_{2}\,{\alpha }_{3}}}+{{k_{6}\, \gamma _{23}}\over {{\alpha }_{2}\,{\alpha }_{3}}}+{{k_{5}\, \gamma _{13}}\over {{\alpha }_{1}\,{\alpha }_{3}}}\\ {{ C}_{45}}= & {} {{k_{4}\,\gamma _{13}}\over {{\alpha }_{1}\, {\alpha }_{2}}}-{{2\,k_{5}\,\gamma _{12}}\over {{\alpha }_{1} \,{\alpha }_{2}}}-{{k_{6}}\over {{\alpha }_{2}}}\\ {{ C}_{46}}= & {} {{k_{4}\,\gamma _{23}}\over {{\alpha }_{1}\, {\alpha }_{2}}}+{{k_{5}}\over {{\alpha }_{1}}}\\ {{ C}_{47}}= & {} -{{2\,k_{1}\,\gamma _{12}\,\gamma _{23}}\over { {\alpha }_{1}\,{\alpha }_{2}}}-{{k_{2}\,\gamma _{23}}\over { {\alpha }_{2}}}-{{k_{1}\,\gamma _{13}}\over {{\alpha }_{1}}}- {{2\,k_{7}\,\gamma _{12}}\over {{\alpha }_{1}\,{\alpha }_{2}}} \\ {{ C}_{50}}= & {} 0\\ {{ C}_{51}}= & {} -{{4\,k_{1}\,\gamma _{12}\,\gamma _{23}}\over {{\alpha }_{1}\,{\alpha }_{2}\,{\alpha }_{3} }}-{{2\,k_{2}\,\gamma _{23}}\over {{\alpha }_{2}\,{\alpha }_{3 }}}-{{2\,k_{1}\,\gamma _{13}}\over {{\alpha }_{1}\,{\alpha }_{ 3}}}-{{k_{7}\,\gamma _{12}}\over {{\alpha }_{1}\,{\alpha }_{2} \,{\alpha }_{3}}}-{{k_{3}}\over {{\alpha }_{3}}}\\ {{ C}_{52}}= & {} {{k_{7}}\over {{\alpha }_{1}\,{\alpha }_{3}}} \\ {{ C}_{53}}= & {} -{{2\,k_{4}\,\gamma _{13}\,\gamma _{23}}\over { {\alpha }_{1}\,{\alpha }_{2}\,{\alpha }_{3}}}-{{k_{7}\, \gamma _{23}}\over {{\alpha }_{1}\,{\alpha }_{2}\, {\alpha }_{3}}}-{{k_{5}\,\gamma _{13}}\over {{\alpha }_{1}\, {\alpha }_{3}}}\\ {{ C}_{54}}= & {} -{{2\,k_{4}\,\gamma _{12}\,\gamma _{23}}\over {{\alpha }_{1}\,{\alpha }_{2}\,{\alpha }_{3} }}-{{2\,k_{4}\,\gamma _{13}}\over {{\alpha }_{1}\,{\alpha }_{3 }}}+{{k_{5}\,\gamma _{12}}\over {{\alpha }_{1}\,{\alpha }_{3} }}+{{k_{6}}\over {{\alpha }_{3}}} \end{aligned}$$

$$\begin{aligned} {{ C}_{55}}= & {} {{4\,k_{1}\, \gamma _{12}\,\gamma _{13}\,\gamma _{23}}\over {{\alpha }_{1}\, {\alpha }_{2}\,{\alpha }_{3}}}+{{2\,k_{4}\,\gamma _{13}\, \gamma _{23}}\over {{\alpha }_{1}\,{\alpha }_{2}\, {\alpha }_{3}}}+{{2\,k_{2}\,\gamma _{13}\,\gamma _{23}}\over { {\alpha }_{2}\,{\alpha }_{3}}}-{{2\,k_{5}\,\gamma _{12}\, \gamma _{23}}\over {{\alpha }_{1}\,{\alpha }_{2}\, {\alpha }_{3}}}-{{k_{6}\,\gamma _{23}}\over {{\alpha }_{2}\, {\alpha }_{3}}}\\&+\, {{2\,k_{1}\,\gamma _{13}^2}\over {{\alpha }_{1 }\,{\alpha }_{3}}}+{{2\,k_{7}\,\gamma _{12}\,\gamma _{13}}\over { {\alpha }_{1}\,{\alpha }_{2}\,{\alpha }_{3}}}+{{k_{3}\, \gamma _{13}}\over {{\alpha }_{3}}}+{{k_{4}\,\gamma _{12}}\over { {\alpha }_{1}\,{\alpha }_{2}}}\\ {{ C}_{56}}= & {} {{2\,k_{4}\, \gamma _{23}^2}\over {{\alpha }_{1}\,{\alpha }_{2}\, {\alpha }_{3}}}+{{k_{5}\,\gamma _{23}}\over {{\alpha }_{1}\, {\alpha }_{3}}}-{{k_{4}}\over {{\alpha }_{1}}}\\ {{ C}_{57}}= & {} -{{4\,k_{1}\,\gamma _{12}\,\gamma _{23}^2}\over { {\alpha }_{1}\,{\alpha }_{2}\,{\alpha }_{3}}}-{{2\,k_{2} \,\gamma _{23}^2}\over {{\alpha }_{2}\,{\alpha }_{3}}}-{{4\,k _{4}\,\gamma _{12}\,\gamma _{13}\,\gamma _{23}}\over {{\alpha }_{1} \,{\alpha }_{2}\,{\alpha }_{3}}}-{{2\,k_{1}\,\gamma _{13}\, \gamma _{23}}\over {{\alpha }_{1}\,{\alpha }_{3}}}-{{4\,k_{7} \,\gamma _{12}\,\gamma _{23}}\over {{\alpha }_{1}\,{\alpha }_{2 }\,{\alpha }_{3}}}\\&-\,{{k_{3}\,\gamma _{23}}\over {{\alpha }_{3} }}-{{2\,k_{5}\,\gamma _{12}\,\gamma _{13}}\over {{\alpha }_{1}\, {\alpha }_{3}}}-{{2\,k_{7}\,\gamma _{13}}\over {{\alpha }_{1} \,{\alpha }_{3}}}-{{k_{1}\,\gamma _{12}}\over {{\alpha }_{1}}} -k_{2}\end{aligned}$$

$$\begin{aligned} {{ C}_{60}}= & {} 0\\ {{ C}_{61}}= & {} -{{k_{7}}\over { {\alpha }_{2}\,{\alpha }_{3}}}\\ {{ C}_{62}}= & {} -{{4\,k_{1}\, \gamma _{12}\,\gamma _{23}}\over {{\alpha }_{1}\,{\alpha }_{2} \,{\alpha }_{3}}}-{{2\,k_{2}\,\gamma _{23}}\over {{\alpha }_{2 }\,{\alpha }_{3}}}-{{2\,k_{1}\,\gamma _{13}}\over {{\alpha }_{ 1}\,{\alpha }_{3}}}+{{k_{7}\,\gamma _{12}}\over {{\alpha }_{1} \,{\alpha }_{2}\,{\alpha }_{3}}}-{{k_{3}}\over {{\alpha } _{3}}}\\ {{ C}_{63}}= & {} {{k_{7}\,\gamma _{13}}\over {{\alpha }_{1} \,{\alpha }_{2}\,{\alpha }_{3}}}\\ {{ C}_{64}}= & {} -{{2\,k_{4} \,\gamma _{23}}\over {{\alpha }_{2}\,{\alpha }_{3}}}-{{2\,k_{4 }\,\gamma _{12}\,\gamma _{13}}\over {{\alpha }_{1}\,{\alpha }_{ 2}\,{\alpha }_{3}}}+{{2\,k_{5}\,\gamma _{12}^2}\over { {\alpha }_{1}\,{\alpha }_{2}\,{\alpha }_{3}}}+{{k_{6}\, \gamma _{12}}\over {{\alpha }_{2}\,{\alpha }_{3}}}-{{k_{5} }\over {{\alpha }_{3}}}\\ {{ C}_{65}}= & {} {{k_{7}\,\gamma _{13} }\over {{\alpha }_{2}\,{\alpha }_{3}}}+{{k_{4}}\over { {\alpha }_{2}}}\\ {{ C}_{66}}= & {} -{{k_{4}\,\gamma _{12}}\over { {\alpha }_{1}\,{\alpha }_{2}}}\\ {{ C}_{67}}= & {} {{4\,k_{1}\, \gamma _{12}\,\gamma _{13}\,\gamma _{23}}\over {{\alpha }_{1}\, {\alpha }_{2}\,{\alpha }_{3}}}-{{2\,k_{4}\,\gamma _{13}\, \gamma _{23}}\over {{\alpha }_{2}\,{\alpha }_{3}}}+{{2\,k_{2} \,\gamma _{13}\,\gamma _{23}}\over {{\alpha }_{2}\,{\alpha }_{3 }}}-{{2\,k_{7}\,\gamma _{23}}\over {{\alpha }_{2}\,{\alpha }_{ 3}}}+{{2\,k_{1}\,\gamma _{13}^2}\over {{\alpha }_{1}\, {\alpha }_{3}}}-\,{{k_{5}\,\gamma _{13}}\over {{\alpha }_{3}}}\\&\, + {{k_{3}\,\gamma _{13}}\over {{\alpha }_{3}}}-{{2\,k_{1}\,\gamma _{ 12}^2}\over {{\alpha }_{1}\,{\alpha }_{2}}}-{{k_{2}\,\gamma _{ 12}}\over {{\alpha }_{2}}}+k_{1}\\ \end{aligned}$$

$$\begin{aligned}&{{ C}_{70}}={{k_{7}}\over { {\alpha }_{1}\,{\alpha }_{2}\,{\alpha }_{3}}}\\&{{ C}_{71}}=0\\&{{ C}_{72}}=0\\&{{ C}_{73}}=0\\&{{ C}_{74}}= {{2\,k_{7}\,\gamma _{12}}\over {{\alpha }_{1}\,{\alpha }_{2}\, {\alpha }_{3}}}\\&{{ C}_{75}}={{2\,k_{4}\,\gamma _{13}^2}\over { {\alpha }_{1}\,{\alpha }_{2}\,{\alpha }_{3}}}-{{2\,k_{5} \,\gamma _{12}\,\gamma _{13}}\over {{\alpha }_{1}\,{\alpha }_{2 }\,{\alpha }_{3}}}+{{k_{7}\,\gamma _{13}}\over {{\alpha }_{1} \,{\alpha }_{2}\,{\alpha }_{3}}}-{{k_{6}\,\gamma _{13}}\over { {\alpha }_{2}\,{\alpha }_{3}}}+{{2\,k_{1}\,\gamma _{12} }\over {{\alpha }_{1}\,{\alpha }_{2}}}+{{k_{2}}\over { {\alpha }_{2}}}\\&{{ C}_{76}}={{2\,k_{4}\,\gamma _{13}\,\gamma _{23}}\over {{\alpha }_{1}\,{\alpha }_{2}\,{\alpha }_{3} }}+{{2\,k_{7}\,\gamma _{23}}\over {{\alpha }_{1}\,{\alpha }_{2 }\,{\alpha }_{3}}}+{{k_{5}\,\gamma _{13}}\over {{\alpha }_{1} \,{\alpha }_{3}}}-{{k_{1}}\over {{\alpha }_{1}}}\\&{{ C}_{77}}={{2\,k_{4}\,\gamma _{13}\,\gamma _{23}}\over { {\alpha }_{1}\,{\alpha }_{2}\,{\alpha }_{3}}}+{{k_{5}\, \gamma _{13}}\over {{\alpha }_{1}\,{\alpha }_{3}}}\\ \end{aligned}$$