The non-polynomial conservation laws and integrability analysis of generalized Riemann type hydrodynamical equations

Based on the gradient-holonomic algorithm we analyse the integrability property of the generalized hydrodynamical Riemann type equation for arbitrary . The infinite hierarchies of polynomial and non-polynomial conservation laws, both dispersive and dispersionless are constructed. Special attention is paid to the cases N = 2, 3 and N = 4, for which the conservation laws, Lax type representations and Hamiltonian structures are analysed in detail. We also show that the case N = 2 is equivalent to a generalized Hunter–Saxton dynamical system, whose integrability follows from the results obtained. As a by-product of our analysis we demonstrate a new set of non-polynomial conservation laws for the related Hunter–Saxton equation.