A Benney-Like Lattice

We propose a new Benney-like lattice and show that the new system of equations can be reduced to Chaplygin gas-like equations as well as the heavenly equation. We construct two infinite sets of conserved charges. The conserved densities are related to Legendre polynomials. We prove that the system is bi-Hamiltonian and that the conserved charges are in involution with respect to either of the Hamiltonian structures. We show that our Lax operator generates a new dispersionless Toda hierarchy.