Invariant Measures of Modified $\mathrm{LR}$ and $\mathrm{L+R}$ Systems

We introduce a class of dynamical systems having an invariant measure, the modifications of well-known systems on Lie groups: $\mathrm{LR}$ and $\mathrm{L+R}$ systems. As an example, we study a modified Veselova nonholonomic rigid body problem, considered as a dynamical system on the product of the Lie algebra $so(n)$ with the Stiefel variety $V_{n, r}$, as well as the associated $\epsilon\mathrm{L+R}$ system on $so(n) \times V_{n, r}$. In the $3$-dimensional case, these systems model the nonholonomic problems of motion of a ball and a rubber ball over a fixed sphere.