  The generalized Chazy differential equation corresponds to the following
two-parameter family of differential equations \begin{equation*}\label{gcdeq}
\dddot x+|x|^q \ddot x+\dfrac{k |x|^q}{x}\dot x^2=0, \end{equation*} which has
its regularity varying with $q$ , a positive integer. Indeed, for $q=1$ it is
discontinuous on the straight line $x=0$, whereas for $q$ a positive even
integer it is polynomial, and for $q>1$ a positive odd integer it is continuous
but not differentiable on the straight line $x=0$. In 1999, the existence of
periodic solutions in the generalized Chazy differential equation was
numerically observed for $q=2$ and $k=3$. In this paper, we prove analytically
the existence of such periodic solutions. Our strategy allows to establish
sufficient conditions ensuring that the generalized Chazy differential
equation, for $k=q+1$ and any positive integer $q$ , has actually an invariant
topological cylinder foliated by periodic solutions in the $(x,\dot x,\ddot
x)$-space. In order to set forth the bases of our approach, we start by
considering $q=1,2,3$, which are representatives of the different classes of
regularity. For an arbitrary positive integer $q$, an algorithm is provided for
checking the sufficient conditions for the existence of such an invariant
cylinder, which we conjecture that always exists. The algorithm was
successfully applied up to $q=100$.
