  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.
