An algebraically stable variety for a four-dimensional dynamical system reduced from the lattice super-KdV equation (Q2658239)

In [J. Math. Phys. 60, No. 9, 093503, 8 p. (2019; Zbl 1422.37044)] the same authors, applying the traveling wave reduction to the lattice super-KdV equation in the case of finitely generated Grassmann algebra, obtained a four-dimensional discrete integrable dynamical system: \[ \varphi: \left\{ \begin{array}{rcl} \bar{x_{0}}&=&x_{2},\\ \bar{x_{1}}&=&x_{3},\\ \bar{x_{2}}&=&-x_{2}-x_{0}+\dfrac{hx_{2}}{1-x_{2}},\\ \bar{x_{3}}&=&-x_{1}-x_{3}+\dfrac{2-x_{2}+hx_{3}}{(1-x_{2})^{2}}. \end{array} \right. \] This system is a Quispel-Roberts-Thompson (QRT) map, a two-dimensional map generating an automorphism of a rational elliptic surface, for variables \(x_0 , x_2\) coupled with linear equations for variables \(x_1 , x_3\) with coefficients depending on \(x_2\). This system has two invariants but does not satisfy the singularity confinement criterion. Furthermore, they observe that the dynamical degree of the above map grows quadratically, which is a rather unusual phenomenon. In this paper, constructing a rational variety where the system is lifted to an algebraically stable map, it is shown that one can find invariants using the action on the Picard group. For the entire collection see [Zbl 1459.00016].