  We generalize the Benney lattice and show that the new system of equations
can be reduced to a generalized Chaplygin gas as well as the heavenly equation.
We construct two infinite sets of conserved charges and show that one of the
sets can be obtained from the Lax function. The conserved densities are related
to Legendre polynomials and we present closed form expressions for the
generating functions for these densities which also determines the Riemann
invariants of the problem. 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 the associated generalized elastic medium
equations are bi-Hamiltonian as well. We also bring out various other
interesting features of our model.
