Periodic orbits and integrability of Rocard's system

In 1941, based on Van der Pol's relaxation oscillator equation, physicist Yves Rocard in the book (Rocard 1941) proposed a relaxation econometric oscillator to describe cyclical oscillations in the economy. Furthermore, it was later found that the model exhibits chaotic phenomenon. Rocard's chaotic system predates Lorenz's discovery by 22 years, which is a three-dimensional autonomous differential system. In this paper, we research periodic orbits and integrability of Rocard's system. We study the zero-Hopf bifurcation near equilibria and center problem on center manifolds, proving that one or three periodic orbits can bifurcate through the application of the averaging method up to arbitrary finite order, while obtaining center conditions for all equilibria through Lyapunov method. Furthermore, we investigate the integrability of Rocard's system, which has no algebraic first integrals, Darboux polynomials, or Darboux first integrals, and is not Liouvillian integrable.