Commuting conserved quantities in nonlinear realizations of W3

We derive commuting conserved quantities for the Boussinesq equations and provide geometrical origin for them in the framework of the nonlinear realizations of W3. We show that these quantities are the translation generators on the coset manifold and their involution properties correspond to the linear independence of the coordinates that parametrize the coset manifold. These quantities form a Cartan subalgebra in the infinite dimensional linear algebra W3∞ which is an enveloping algebra of W3. 15 refs