Invariant measures of modified LR and L+R systems

We introduce a class of dynamical systems having an invariant measure, the modifications of well known systems on Lie groups: LR and L+R systems. As an example, we study 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 $ϵ$L+R system on $so(n)\times V_{n,r}$. In the 3–dimensional case, these systems model the nonholonomic problems of a motion of a ball and a rubber ball over a fixed sphere.