Measure preservation and integrals for Lotka--Volterra $T$-systems and their Kahan discretisation

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.