Reductions of the strict KP hierarchy

Let $R$ be a commutative complex algebra and $\partial$ be a $\mathbb{C}$-linear derivation of $R$ such that all powers of $\partial$ are $R$-linearly independent. Let $R[\partial]$ be the algebra of differential operators in $\partial$ with coefficients in $R$ and $Psd$ be its extension by the pseudodifferential operators in $\partial$ with coefficients in $R$. In the algebra $R[\partial]$, we seek monic differential operators $\mathbf{M}_n$ of order $n\ge2$ 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 $M=(\mathbf{M}_n)^{1/n}$ satisfy zero-curvature conditions, which suffice for obtaining the Lax equations for $\mathbf{M}_n$ and, in particular, for proving that the $n$th root $M$ of $\mathbf{M}_n$ is a solution of the strict KP theory if and only if $\mathbf{M}_n$ 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.