  Based on the gradient-holonomic algorithm we analyze the integrability
property of the generalized hydrodynamical Riemann type equation
$%D_{t}^{N}u=0$ for arbitrary $N\in \mathbb{Z}_{+}.$ 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
bi-Hamiltonian structures are analyzed 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 byproduct of our analysis
we demonstrate a new set of non-polynomial conservation laws for the related
Hunter-Saxton equation.
