  Symmetries and conservation laws are studied for two classes of physically
and analytically interesting radial wave equations with power nonlinearities in
multi-dimensions. The results consist of two main classifications: all
symmetries of point type and all conservation laws of a general energy-momentum
type are explicitly determined, including those such as dilations, inversions,
similarity energies and conformal energies that exist only for special powers
or dimensions. In particular, all variational cases (when a Lagrangian
formulation exists) and non-variational cases (when no Lagrangian exists) for
these wave equations are considered. As main results, the classification yields
generalized energies and radial momenta in certain non-variational cases, which
are shown to arise from a new type of Morawetz dilation identity that produces
conservation laws for each of the two wave equations in a different way than
Noether's theorem.
