Commuting Ordinary Differential Operators with Polynomial Coefficients and Automorphisms of the First Weyl Algebra

In this paper, we study rank 2 commuting ordinary differential operators with polynomial coefficients and the orbit space of the automorphisms group of the first Weyl algebra on such operators. We prove that for arbitrary fixed spectral curve of genus one the space of orbits is infinite. Moreover, we prove in this case that for any n >= 1 there is a pair of self-adjoint commuting ordinary differential operators of rank 2 L-4 = (partial derivative(2)(x) + V(x))(2) + W(x), L-6, where W(x), V(x) are polynomials of degree nand n+2. We also prove that there are hyperelliptic spectral curves with the infinite spaces of orbits.