Commuting ordinary differential operators with polynomial coefficients

Construction of explicit examples of commuting differential operators is a very complicated problem, especially for operators with polynomial coefficients. There are known a few such examples now. I will tell about some recent results obtained at the junction of analytic and algebraic theories of commuting differential operators and concerning Berest's conjecture on orbits of the group of automorphisms of the Weyl algebra on the solution space of an algebraic equation.