Non-abelian Painlevé systems with generalized Okamoto integral

We study non-abelian systems of Painlevé type. To derive them, we introduce an auxiliary autonomous system with the frozen independent variable and postulate its integrability in the sense of the existence of a non-abelian first integral that generalizes the Okamoto Hamiltonian. All non-abelian P6-P2-systems with such integrals are found. A coalescence limiting scheme is constructed for these non-abelian Painlevé systems. This allows us to construct an isomonodromic Lax pair for each of them.