Let $g=\sqrt{adx^2+bdxdy+cdy^2}$ be some Riemannian metric, where $a,b,c$ are positive constant numbers. That is to say $$(g)=\begin{pmatrix}a&b/2\\ \:\:\:b/2&c\end{pmatrix}. $$ Assume that $\mathcal{C}$ is the geometric circle associate to the Riemannian metric $g$, that is $\mathcal{C}$ is given by the folllwoing equation: $$\begin{pmatrix}x&y\end{pmatrix}\begin{pmatrix}a&b/2\\ \:\:\:b/2&c\end{pmatrix}\begin{pmatrix}x\\ y\end{pmatrix}=1. $$
My question is that what kinf of symmetry has $\mathcal{C}$, as $g$ is symmetric with respect to the diagonal?