periodic without being $n$-periodic if and only if there exists a pair of real polynomials $p_{d_0}$, $q_{d_0}$ of degrees $d_0$, $d_0$ respectively, and satisfying the following:  -   ${\pazocal{C}}_{\gamma}$ is an ellipse, $0<\gamma<a$, and      -   $n=0m$ is even, $d_0=d_0=m-0$, $$s\left(s