Periodic solutions of second-order differential equations with two-dimensional Lie point symmetry algebra

In this paper, we study some aspects of the dynamics in the phase plane of smooth second-order differential equations ẍ=w(x,ẋ) possessing an rr-dimensional Lie point symmetry algebra LrLr with r≥2r≥2, focusing on the existence, nonexistence and localization periodic orbits. Finally, it is proved that the polynomial Liénard systems ẍ=f(x)ẋ+g(x) with f,g∈R[x]f,g∈R[x] having an LrLr with r≥2r≥2 do not have limit cycles. As far as we know, this is the first work that relates Lie point symmetries and periodic orbits.