$$  Each real solution $\gamma$ for the above equations for some fixed values of $a$ and $b$ will produce a $0$-elliptic periodic trajectory which is not $0$-periodic. Some examples are shown in Figure \[fig:0-elliptic-periodic\].  (-0.0,-0.0) rectangle (0.0,0.0);  (0,0) circle \[x radius=[sqrt(0)]{}, y radius=[sqrt(0)]{}\];  (0,0) circle \[x radius=[sqrt(0-0.0000)]{}, y radius=[sqrt(0+0.0000)]{}\];  ([sqrt(0)+0]{},-0) - (-0,[sqrt(0)+0]{}); ([-sqrt(0)-0]{},0) - (0,[-sqrt(0)-0]{}); (0,[sqrt(0)+0]{}) - ([-sqrt(0)-0]{},-0); (-0,[-sqrt(0)-0]{}) - ([sqrt(0)+0]{},0);  (0, -0.000000) - (