The matrix analogs of the first Painlevé equations

Earlier, Balandin and Sokolov obtained matrix analogs of the first and the second transcendent Painlevé equations and studied them for possession of the Painlevé property. In the present paper the integrability of the generalizations of the first Painlevé equation are studied using Painlevé–Kowalevskaya test. The main result obtained is integrability sufficient conditions for the generalized matrix analogs of the first Painlevé equation. An important role in finding these criteria is played by decomposition of the matrix into blocks. The obtained results are in agreement with the earlier investigations of special cases of our equations.