Ufa Mathematical Journal
    Volume 3, Number 4, pp. 20-26

    The matrix analogs of the first Painlev\'{e} equations

    Balandin S.P., Cherdantzev I.Yu.

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    Earlier, Balandin and Sokolov obtained matrix analogs of the first and the second transcendent Painlev\'{e} equations and studied them for possession of the Painlev\'{e} property. In the present paper the integrability of the generalizations of the first Painlev\'{e} equation are studied using Painlev\'{e}–Kowalevskaya test. The main result obtained is integrability sufficient conditions for the generalized matrix analogs of the first Painlev\'{e} 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.