Self-adjoint commuting ordinary differential operators of rank two

Self-adjoint commuting ordinary differential operators of rank two are considered. We find sufficient conditions when an operator of fourth order commuting with an operator of order $4g+2$ is self-adjoint. An equation on potentials $V(x),W(x)$ of the self-adjoint operator $L=(\partial _{x}^{2}+V)^{2}+W$ and some additional data is introduced. With the help of this equation examples are constructed.