Gibbons–Tsarev Type Systems and Eventual Identities

We show that non-diagonalizable reductions of the dKP equation associated with regular non-semisimple (Formula presented.) -manifolds cannot exist. The proof is based on the derivation and study of a generalized Gibbons–Tsarev system (gGT system) in the non-semisimple/non-diagonalizable setting. Remarkably, a class of solutions of the gGT system is defined by eventual identities of the underlying regular (Formula presented.) -manifold structure. Furthermore, we use these vector fields to construct integrable reductions of Pavlov's hydrodynamic chain. In this case, the corresponding solutions are defined for any choice of Jordan block structure of the operator of multiplication by an eventual identity.