  We highlight a non-canonical yet natural choice of variables for an efficient
derivation of a kinetic equation for the energy density in non-isotropic
systems, including internal gravity waves on a vertical plane, inertial and
Rossby waves. The existence of a second quadratic invariant simplifies the
kinetic equation and leads to extra conservation laws for resonant
interactions. We analytically determine the scaling of the radial turbulent
energy spectrum. Our findings suggest the existence of an inverse energy
cascade of internal gravity waves, from small to large scales, in practically
relevant scenarios.
