First integrals of adiabatic stellar oscillations

We consider the equations of linear adiabatic oscillations of stars analytically. There are two known first integrals of nonradial oscillations of spherically symmetric stars, which exist for any stellar structure. One is obtained by integrating the linearized Poisson equation in the case of radial oscillations. The other is derived from momentum conservation in the case of dipolar oscillations. Both of these integrals reduce the order of the differential equations from four to two, which is not trivial. We find that this property can be understood from the fact that the problem of adiabatic stellar oscillations can be formulated as a Hamiltonian system. We then examine the symmetry of the system that corresponds to these first integrals. We finally show that there is no other first integral that is valid for arbitrary structures of stars.