We show that any Lotka--Volterra $T$-system associated with an $n$-vertex tree $T$ as introduced in Quispel et al., J. Phys. A 56 (2023) 315201, preserves a rational measure. We also prove that the Kahan discretisation of these $T$-systems factorises and preserves the same measure. As a consequence, for the Kahan maps of Lotka--Volterra systems related to the subclass of $T$-systems corresponding to graphs with more than one $n$-vertex subtree, we are able to construct rational integrals.

翻译：我们证明，与Quispel等人（J. Phys. A 56 (2023) 315201）引入的 $n$ 顶点树 $T$ 相关联的任何洛特卡-沃尔泰拉 $T$ 系统均保持一个有理测度。同时，我们证明这些 $T$ 系统的卡汉离散化可作因式分解，并保持相同测度。由此，对于与包含多个 $n$ 顶点子树的图对应的 $T$ 系统子类相关的洛特卡-沃尔泰拉系统的卡汉映射，我们能够构造有理积分。