We show how to construct in an elementary way the invariant of the KHK discretisation of a cubic Hamiltonian system in two dimensions. That is, we show that this invariant is expressible as the product of the ratios of affine polynomials defining the prolongation of the three parallel sides of a hexagon. On the vertices of such a hexagon lie the indeterminacy points of the KHK map. This result is obtained analysing the structure of the singular fibres of the known invariant. We apply this construction to several examples, and we prove that a similar result holds true for a case outside the hypotheses of the main theorem, leading us to conjecture that further extensions are possible.

翻译：我们展示了如何以初等方式构造二维三次哈密顿系统KHK离散化的不变量。具体而言，我们证明了该不变量可表示为定义六边形三条平行边延拓的仿射多项式之比的乘积，而该六边形的顶点恰为KHK映射的不确定点。这一结果是通过分析已知不变量奇异纤维结构得出的。我们将此构造应用于多个实例，并证明在偏离主定理假设的案例中仍存在类似结论，由此推测该构造可进一步推广。