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。即证明该不变量可表示为定义六边形三条平行边延长线的仿射多项式比值之积，而该六边形的顶点恰为KHK映射的不定点。该结果通过分析已知不变量的奇异纤维结构获得。我们将此构造应用于多个实例，并证明在主定理假设之外的情形中仍存在类似结果，由此推测该构造可进一步推广。