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

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