Hamiltonian reductions in Matrix Painlev\'e systems

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