For certain finite groups G of Bäcklund transformations, we show that a dynamics of G-invariant configurations of n| G| Calogero–Painlevé particles is equivalent to a certain n-particle Calogero–Painlevé system. We also show that the reduction of a dynamics on G-invariant subset of n| G| × n| G| matrix Painlevé system is equivalent to a certain n× n matrix Painlevé system. The groups G correspond to folding transformations of Painlevé equations. The proofs are based on Hamiltonian reductions.
