Solution
= Solution
Choose unequal positive numbers $a,b$. The planar <ellipse>
$$
\gamma(t)=(a\cos t,b\sin t,0),
\qquad 0\leq t\leq2\pi,
$$
is a simple closed curve and is not a circle. The nonidentity proper <rigid motion>
$$
R=\operatorname{diag}(-1,-1,1),
\qquad T=0,
$$
is rotation through $\pi$ about the $z$-axis, and $R\gamma(t)=\gamma(t+\pi)$. It therefore preserves the curve as a set.