Hamiltonian reductions in matrix Painlevé systems

For certain finite groups $G$ of B\"acklund transformations we show that the dynamics $G$-invariant configurations $n|G|$ Calogero--Painlev\'e particles is equivalent to $n$-particle system. We also reduction on subset $n|G|\times n|G|$ matrix Painlev\'e system $n\times n$ The correspond folding equations. proofs are based Hamiltonian reductions.