How one can repair non-integrable Kahan discretizations. II. A planar system with invariant curves of degree 6

We find a novel one-parameter family of integrable quadratic Cremona maps of the plane preserving a pencil of curves of degree 6 and of genus 1. They turn out to serve as Kahan-type discretizations of a novel family of quadratic vector fields possessing a polynomial integral of degree 6 whose level curves are of genus 1, as well. These vector fields are non-homogeneous generalizations of reduced Nahm systems for magnetic monopoles with icosahedral symmetry, introduced by Hitchin, Manton and Murray. The straightforward Kahan discretization of these novel non-homogeneous systems is non-integrable. However, this drawback is repaired by introducing adjustments of order $O(\epsilon^2)$ in the coefficients of the discretization, where $\epsilon$ is the stepsize.


INTRODUCTION
The problem of integrable discretization [29] consists of finding, for a given integrable system, a discretization which remains integrable. All conventional discretization methods for ODEs, like Runge-Kutta methods etc., fail to preserve integrability. However, there exists an "unconventional" numerical method applicable to any system of ODEs on R n with a quadratic vector field, known as Kahan discretization, which possesses remarkable properties in this respect. Consider a quadratic ODĖ where Q : R n → R n is a vector of quadratic forms, B is an n × n matrix, and c ∈ R n . Kahan discretization, introduced in [14], consists in replacing the time derivative on the left-hand side by the first difference of the numerical approximation x : Z → R n , while the quadratic expressions on the right-hand side are replaced by symmetric bilinear expressions in terms of x = x(t) and x = x(t + ): where is the symmetric bilinear form corresponding to the quadratic form Q. Equation (2) is linear with respect to x and therefore defines a rational map x = f (x, ). Due to the symmetry of equation (2) with respect to interchanging x ↔ x accompanied by sign inversion → − , the map f is reversible: Thus, the map f is birational. For some reasons which remain not completely clarified up to now, Kahan's method tends to preserve integrability much more often than any other known general purpose discretization scheme. 1 This was first observed by Hirota and Kimura [10,15], who (being unaware of the work by Kahan) applied this scheme to the Euler top and to the Lagrange top, and observed that the resulting maps are integrable. Since then, integrability properties of Kahan's method when applied to integrable systems (also called "Hirota-Kimura method" in this context) were extensively studied, see [16-23, 25, 26, 30] and [2-5, 12, 13]. Integrability is preserved in an amazing number of cases, but not always.
If (γ 1 , γ 2 , γ 3 ) = (1, 1, 2), one can find non-homogeneous perturbations of the quartic polynomial H(x, y) so that the resulting differential equations (4) still have the above mentioned property: all integral curves are of genus 1. A Kahan discretization of the perturbed (non-homogeneous) system is non-integrable. However, it was shown in [26] that one can adjust the coefficients of the discretization (making them dependent on in a non-trivial way) to obtain an integrable Kahan-type discretization.
The present paper is devoted to a similar result for systems of the class (γ 1 , γ 2 , γ 3 ) = (1, 2, 3). The homogeneous system can be taken as It possesses an integral of motion of degree 6: whose level sets are curves of genus 1. The Kahan discretization of this system reads: It is integrable, with an integral of motion Consider the following non-homogeneous perturbation of system (5): It has the following integral of motion: with the same property as above (all level sets are curves of genus 1). The Kahan discretization of this system, generates a non-integrable map. However, the coefficients of this discretization can be adjusted via O( 2 ) terms, to produce an integrable map: This map, like the unperturbed one (7), has an integral of motion whose level sets are curves of degree 6 and of genus 1 (the irreducible ones). The presentation is organized as follows. In Section 2, we consider in detail system (5) and its Kahan discretization (7), paying special attention to the singularity confinement property of the latter map. In Section 3, we perform, following [24], a reduction of the pencil of invariant curves of degree 6 of the Kahan discretization to a pencil of biquadratic curves. This way, the map is shown to be birationally equaivalent to a special QRT root (cf. [9,27]). In Section 4, we show that the relevant geometric and dynamical properties of this QRT root can be found in a one-parameter family of such maps, and then find a corresponding one-parameter family of birationally equivalent Kahan-type maps preserving a pencil of curves of degree 6 and of genus 1. Finally, in Section 5, a continuous limit is performed in those Kahan-type maps, leading to a novel integrable system (9), with a pencil of invariant curves with the same property (level sets of the non-homogeneous sextic polynomial (10) are of genus 1).
The general case is obtained from this by the re-scaling (x, y) → ( x, y). A simple computation gives an explicit formula for the map f : In homogeneous coordinates, In the following proposition, we collect the relevant information about this map, as found in [4,17,25,30]. (14) admits an integral of motion

Proposition 1. The map f given in
of deg = 6 possesses eleven (distinct) base points given by: • six finite base points of multiplicity 1 on the line 1 = 0: • three base points of multiplicity 2 on the line 2 = 0, two finite and one at infinity: • and two finite base points of multiplicity 3 on the line 3 = 0: See Fig. 1 for an illustration. One has: I( f ) = {p 6 , p 9 , p 11 } and I( f −1 ) = {p 1 , p 7 , p 10 }. All base points participate in three confined singular orbits of the map f : We refer the reader to [1,[6][7][8] for general information about birational (Cremona) maps of CP 2 , including the notion of confined singular orbits (related to degree-lowering curves and dynamical degree, or algebraic entropy).

REDUCTION OF THE MAP f TO A SPECIAL QRT ROOT
We use notation E 6 = P (6; p 1 , . . . , p 6 , p 2 7 , p 2 8 , p 2 9 , p 3 10 , p 3 11 ) (22) for the pencil of curves of degree 6 with simple base boints p 1 , . . . , p 6 , double base points p 7 , p 8 , p 9 , and triple base points p 10 , p 11 . One can simplify such a pencil by applying a quadratic Cremona map φ with the fundamental points p 9 , p 10 , p 11 (both the triple base points and one of the double base points), cf. [24]. Proposition 2. Consider a quadratic Cremona map φ blowing down the lines (p 10 p 11 ), (p 9 p 11 ), (p 9 p 10 ) to points denoted by q 9 , q 10 , q 11 , respectively, and blowing up the points p 9 , p 10 , p 11 to the lines (q 10 q 11 ), (q 9 q 11 ), (q 9 q 10 ). All other base points p i , i = 1, . . . , 8 are regular points of φ and their images are denoted by q i = φ(p i ). The change of variables φ maps pencil (22) of sextic curves to the pencil E 4 = P (4; q 1 , . . . , q 6 , q 10 , q 11 , q 2 7 , q 2 8 ) (23) of quartic curves with eight simple base points and two double base points. The point q 9 is not a base point of the latter pencil.
Proof. The total image of a curve C ∈ E 6 is a curve of degree 12. Since C passes through p 9 , p 10 , p 11 with multiplicities 2,3,3, its total image contains the lines (q 10 q 11 ), (q 9 q 11 ), (q 9 q 10 ) with the same multiplicities. Dividing by the linear defining polynomials of all these lines, we see that the proper image of C is a curve of degree 12 − 8 = 4. This curve passes through all points q i , i = 1, . . . , 8 (for i = 7, 8 with multiplicity 2). The curve C of degree 6 has no other intersections with the line (p 10 p 11 ) different from two triple points p 10 and p 11 , therefore its proper image does not pass through q 9 . On the other hand, the curve C of degree 6 has one additional intersection point with each of the lines (p 9 p 10 ) and (p 9 p 11 ), different from the double point p 9 and the triple point p 10 , respectively p 11 . Therefore, its proper image passes through q 11 , resp. q 10 , with multiplicity 1.
For the proof of the following Proposition, we will repeatedly use the following lemma. Proof. The total image of the line (ad) is a conic, but since a is blown up to a line, the proper image is a line. This line has to pass through D = F(d) and through A (since the line (ad) intersects the line (bc) which is blown down to A).
The point q 9 is its fixed point, and lies on the line (q 7 q 8 ). Moreover, the points q 3 and q 11 are infinitely near.
Proof. We have: 3 for f , then for φ).
Next, we consider lines which are blown down by g: (indeed, the total f -image of the conic is a curve of degree 4; however, three lines split off, being the blow-ups of p 6 , p 9 , p 11 ; thus, the proper image is the line through f (p 8 ) = p 9 and f (p 10 ) = p 11 ); (q 6 q 11 ) (applying Lemma 3 for φ −1 , then for f ). The fact that q 3 and q 11 are infinitely near follows from the fact that p 3 ∈ (p 9 p 10 ), the latter line being blown down to q 11 by φ.
It remains to show that q 9 ∈ (q 7 q 8 ). For this, observe that the total φ-image of (p 7 p 8 ) is the conic C(q 7 , q 8 , q 9 , q 10 , q 11 ). However, since p 9 ∈ (p 7 p 8 ), the blow-up of p 9 splits off this conic. This is the line (q 10 q 11 ), and it does not contain any of the points q 7 , q 8 , q 9 . Thus, the proper φ-image of (p 7 p 8 ) is a line containing the latter three points, which are therefore collinear.

GENERALIZATION OF THE QRT ROOT
We try to generalize the map of the previous section. All objects found here will be one-parameter perturbations (with the parameter c) of the corresponding objects from the previous section. We will refrain from indicating this by an extra c in the notation (to keep it as brief as possible). However, the reader should keep in mind that the unperturbed situation corresponds to c = 0.
The idea is to stay in the class of symmetric QRT roots of deg = 2: in non-homogeneous coordinates, g(u, v, 1) = [ u : v : 1] with which admit an integral of motion Note that map (26) corresponds to α = 1/2, β = −1/2. As a characteristic feature we choose the existence of a short singular orbit (the third one in (24)): i.e., of a point q 11 which belongs both to I(g) and to I(g −1 ). One easily computes: We have a one-parameter generalization of the previous case, with q 11 = (−1, −1) ∈ I(g) ∩ I(g −1 ), under the condition In what follows, we parametrize the coefficients α, β according to Proposition 5. Under condition (31), the map g given in (26) has three confined singular orbits as in (24). Moreover, the point q 3 is infinitely near to q 11 (with the slope −1). The map g has a fixed point The pencil of invariant curves {K(u, v) = λ} of the map g is as in (23). The eight finite base points q 1 , . . . , q 6 , q 10 , q 11 lie on the conic given by the numerator of K(u, v).
Proof. The second singular orbit in (24) is confirmed by an easy computation. Let us compute the first (long) singular orbit, starting with the remaining point from I(g −1 ), that is, with We compute: One easily computes also that g −1 blows up the point q 10 to the line (q 6 q 8 ), while g blows up the point q 6 to the line (q 7 q 10 ). The fixed point q 9 is given by a straightforward computation (note that for c = 0, the point q 9 does not lie on (q 7 q 8 ), the line at infinity).
All this is illustrated on Fig. 3.
Proof. The total image of a curve C of the pencil (23) is a curve of degree 8. Since C passes through q 10 , q 11 , its total image contains the lines (p 9 p 11 ), (p 9 p 10 ). Dividing by the linear defining polynomials of these two lines, we see that the proper image of C is a curve of degree 6. This curve passes through all points p i , i = 1, . . . , 8 (for i = 7, 8 with multiplicity 2). The curve C of degree 4 intersects the line (q 10 q 11 ) at two points q 10 , q 11 , and two further points, therefore its proper image passes through p 9 with multiplicity 2. On the other hand, the curve C of degree 4 has three additional intersection points with each of the lines (q 9 q 10 ) and (q 9 q 11 ), different from the points q 10 , respectively q 11 . Therefore, its proper image passes through p 11 , resp. p 10 , with multiplicity 3.
It remains to conjugate the QRT root g by the quadratic change of variables φ −1 .

Proposition 7.
The map f = φ −1 • g • φ is a quadratic Cremona map with three confined singular orbits, as in (21). The eight base points p i , i = 1, . . . , 6, 10, 11 lie on a conic. Proof. We have: Further, (apply Lemma 3 first for g, then for φ −1 ); (apply Lemma 3 for g, taking into account that q 9 is a fixed point); (apply Lemma 3 first for g, taking into account that q 9 is a fixed point and q 1 = g(q 10 ); then, the total φ −1 -image of the conic is a curve of degree 4; however, three lines split off, being the blow-ups of q 9 , q 10 , q 11 ; thus, the proper image is the line through φ −1 (q 1 ) = p 1 and φ −1 (q 7 ) = p 7 ).
Next, we consider lines which are blown down by f : (apply Lemma 3 first for φ, then for g, taking into account that q 9 is a fixed point); (apply Lemma 3 for φ); It remains to show that the points p 1 , . . . , p 6 , p 10 , p 11 lie on a conic. For this, we observe that the total φ −1 -image of the conic C through q 1 , . . . , q 6 , q 10 , q 11 is a curve of deg = 4, from which two lines split off (blow-ups of q 10 , q 11 ). Thus, the proper image is a conic. This conic contains p 1 = φ −1 (q 1 ), . . . , p 6 = φ −1 (q 6 ). It also contains p 10 and p 11 as the consequence of the fact that C has additional intersection points with both blown-down lines (q 9 q 11 ) and (q 9 q 10 ), apart from q 11 and q 10 , respectively.
Proof. A straightforward symbolic computation.
On Fig. 4 one can see several invariant curves {H 1 (x, y) = λ} of the map f .

CONTINUOUS LIMIT
Re-scaling (x, y) → ( x, y) and c → 2 c, we arrive at system (12), which in the limit → 0 is a discretization of system (9). The latter can be written as with H(x, y) = (xy + c) 2 − 2 3 xy + 1 2 This is a one-parameter (inhomogeneous) perturbation of system (5). Like for the unperturbed system, all level sets {H(x, y) = λ} of the integral of motion (45) are curves of genus 1 (and of degree 6). Thus, map (12) is a non-trivial integrable Kahan-type discretization of (9). Integrability of map (12) is in a contrast to non-integrability of the straightforward Kahan discretization (11) of (9). Proposition 9. The map f generated by bilinear equations (11) is non-integrable, in the sense that its singularities are not confined.
Proof. To show this, we restrict ourselves to the case = 1. The resulting quadratic Cremona map has three singularities, p + = (1, 0) and two further points not lying on the line {y = 0}. Likewise, the inverse map has three singularities, p − = (−1, 0) and two further points not lying on the line {y = 0}. Observe that the line {y = 0} is invariant. Thus, for the singularities to be confined, we need that some f n (p − ) = p + for some n ∈ N. The restriction of the map to the line {y = 0} is given by x − x = −2x x + c, or x = ϕ(x) = (x + c)/(2x + 1). One easily sees that, for a generic c, the orbit of x = −1 under this Möbius transformation does not hit x = 1. Indeed, ϕ n (−1) = 1 is a polynomial equation of degree n for c. Thus, for all c but a countable set this equation is not satisfied for any n ∈ N.

CONCLUSIONS
The results of the present paper confirm that the phenomenon discovered and described in [26] is not isolated, namely that in case of non-integrability of the standard Kahan discretization (when applied to an integrable system), its coefficients can be adjusted to restore integrability. Recall that the definition of Kahan's discretization includes a very straightforward dependence on the small stepsize . Namely, it only appears in the denominator of the differences ( x − x)/ which approximate the derivativesẋ, compare (1) and (2). On the contrary, coefficients of the bilinear expressions on the right hand side of (2) are traditionally taken to literally coincide with the coefficients of the quadratic vector fields on the right hand side of (1). This discretization method preserves integrability much more frequently than one would expect a priori, but not always. Our examples show that, if the straightforward recipe fails to preserve integrability, certain adjustments of the coefficients by quantities of the magnitude O( 2 ) may allow to restore integrability. Further extending the list of examples and finding their systematic explanation in terms of addition laws on Abelian varieties remains an important and entertaining task for the future.
This work was done in the frame of a summer research project of MS and YT at Technische Universität Berlin in the Summer-Fall 2020 (which, due to the pandemic, was performed online). Research of YS is supported by the DFG Collaborative Research Center TRR 109 "Discretization in Geometry and Dynamics".