tic} ========================================================  ### 0-elliptic periodic trajectories {#elliptic-periodic-trajectories .unnumbered}  There is a $0$-elliptic periodic trajectory without being $0$-periodic of the billiard within [eq:ellipse]{}, with a non-degenerate caustic ${\pazocal{C}}_{\gamma}$ if and only if, according to Theorem \[th:elliptic-cayley\] one of the following is satisfied:  -   the caustic is an ellipse, with $\gamma \in (0,a)$ and $D_0