Strict self-adjointness and shallow water models

We consider a class of third order equations from the point of view of strict self-adjointness. Necessary and sufficient conditions to the investigated class be strictly self-adjoint are obtained. Then, from a strictly self-adjoint subclass we consider those who admits a suitable scaling transformation. Consequently it is derived a family of equations including the Benjamin-Bona-Mahony, Camassa-Holm and Novikov equation. By a suitable choice of the parameters, we deduce an one-parameter family of equations unifying the last two mentioned equations. Then, using some recent techniques for constructing conserved vectors, we show that from the scale invariance it is obtained as a conserved density the same quantity employed to construct one of the well known Hamiltonians for the cited integrable equations.