On skew-symmetric and general deformations of Lax pseudodifferential operators

A nonlinear deformation is conjectured for the reduction of the third KP flow on the subspace of skew-symmetric operators, and the conjecture is proved for the linearized flow. As a by-product, we find a peculiar (nonquantum) polynomial deformation of the numbers $\left\{\binom{2n+1}{2s+1}\frac{4^{s+1}-1}{s+1}B_{2s+2}\right\}$, where $B_m$'s are the Bernoulli numbers. General open questions and generalizations are also discussed. The conjecture is extended to all the flows, and its linearized version is proved.