Reductions of the strict KP hierarchy

Let R be a commutative complex algebra and ∂ be a C-linear derivation of R such that all powers of ∂ are R-linearly independent. Let R[∂] be the algebra of differential operators in ∂ with coefficients in R and Psd be its extension by the pseudodifferential operators in ∂ with coefficients in R. In the algebra R[∂], we seek monic differential operators Mn of order n ≥ 2 without a constant term satisfying a system of Lax equations determined by the decomposition of Psd into a direct sum of two Lie algebras that lies at the basis of the strict KP hierarchy. Because this set of Lax equations is an analogue for this decomposition of the n-KdV hierarchy, we call it the strict n-KdV hierarchy. The system has a minimal realization, which allows showing that it has homogeneity properties. Moreover, we show that the system is compatible, i.e., the strict differential parts of the powers of M = (Mn)1/n satisfy zero-curvature conditions, which suffice for obtaining the Lax equations for Mn and, in particular, for proving that the nth root M of Mn is a solution of the strict KP theory if and only if Mn is a solution of the strict n-KdV hierarchy. We characterize the place of solutions of the strict n-KdV hierarchy among previously known solutions of the strict KP hierarchy.