Hamiltonian reductions in matrix Painlevé systems

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.