  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.
