which commute with the flow defined by (0.1) (we set τ1 = z and τ2 = τ). The important fact is that the right hand sides Pn of equations (0.2) are polynomials in v and its derivatives, i.e., differential polynomials in v. This fact allows us to treat the KdV hierarchy (0.2) in the following way. Consider the ring R = C[v]n≥0 of polynomials in the variables v (n) (we shall write v for v(0)). Let...