  In this paper we study self-adjoint commuting ordinary differential
operators. We find sufficient conditions when an operator of fourth order
commuting with an operator of order $4g+2$ is self-adjoint. We introduce an
equation on coefficients of the self-adjoint operator of order four and some
additional data. With the help of this equation we find the first example of
commuting differential operators of rank two corresponding to a spectral curve
of arbitrary genus. These operators have polynomial coefficients and define
commutative subalgebras of the first Weyl algebra.
