Ufa Mathematical Journal
    Volume 3, Number 4, pp. 3-6

    The commutation theory in the main meaning.

    Akbasheva M.S., Shabat A.B.

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    In the paper we demonstrate how one can use Poisson bracket in order to build up and to classify commutating pairs of partial differential operators with two independent variables. For considered operators the commutativity condition is reduced to the simple functional equation with shifts of the arguments. The Poisson bracket represents the limiting case of that functional equation in which the shifts are replaced by corresponding directional derivatives.