We show that infinite sequence of conserved quantities and bi-Hamiltonian structure of DLW hierarchy of integrable models are related to the non-Noether symmetry of dispersiveless water wave system.

Symmetries play essential role in dynamical systems, because they usually simplify analysis of evolution equations and often provide quite elegant solution of problems that otherwise would be difficult to handle. In the present paper we show how knowing just single generator of non-Noether symmetry one can construct infinite involutive sequence of conserved quantities and bi-Hamiltonian structure of one of the remarkable integrable models — dispersiveless long wave system. In fact among nonlinear partial differential equations that describe propagation of waves in shallow water there are many interesting integrable models. And most of them seem to have non-Noether symmetries leading to the infinite sequence of conservation laws and bi-Hamiltonian realization of these equations. In dispersiveless long wave system such a symmetry appears to be local, that in some sense simplifies and investigation of its properties and calculations of conserved quantities.

Evolution of dispersiveless long wave system is governed by the following set of
nolinear partial differential equations
^{2})^{2})
^{2} + vw^{2})^{2} + 4v + 2x(ww^{3})

Before we proceed let us note that dispersive water wave system is actually
infinite dimensional Hamiltonian dynamical system. Assuming that ^{2} + v^{2})dx

Now let us pay attention to conservation laws. By integrating third equation of
dispersive water wave system ^{(0)} = ^{(0)}^{(0)} = ^{(1)} = L^{(0)} = 2 ^{(2)} = L^{(1)} = (L^{2}J^{(0)} = 4 ^{(3)} = L^{(2)} = (L^{3}J^{(0)} = 12 ^{2} + v^{2})dx^{(4)} = L^{(3)} = (L^{4}J^{(0)} = 48 ^{2}w + vw^{3})dx^{(n)} = L^{(n − 1)} = (L^{n}J^{(0)}

Author thanks organizers of 11th Regional Conference on Mathematical Physics for kind hospitality. This work was supported by INTAS (00-00561).