The paper deals with a model proposed by F. J. Dyson [J. Math. Mech. 18, 91-101 (1968; Zbl 0197.24501)] which describes a freely expanding gas cloud of tri-axial ellipsoidal shape. The main result is that when the fluid is a monatomic gas with adiabatic index \(\gamma= 5/3\), then the model can be integrated by quadratures. The author points out that the Dyson model possesses Painlevé property with independent variable \(u= \int T dt\) (called thermacy), which entails integrability of the system.