Spinning gas clouds: III. Solutions of minimal energy with precession

We consider the model of rotating and expanding gas cloud originally proposed by Ovsiannikov (1956 Dokl. Akad. Nauk SSSR 111 47) and Dyson (1968 J. Math. Mech. 18 91). Under the restricting assumptions of an adiabatic index γ = 5/3 and of vorticity-free motion, this has been shown (Gaffet 2001 J. Phys. A: Math. Gen. 34 2097) to be a Liouville integrable Hamiltonian system. In the present work, we consider the precessing solutions where the cloud does not retain a fixed rotation axis. Choosing for definiteness a particular set of constants of motion (which corresponds to a minimum of the energy), we show that a separation of variables occurs, and that the equations of motion are reducible to the form of a Riccati equation, whose integration merely involves an elliptic integral