= Solution
The angular <Euler-Lagrange equations> are
$$
I_1\ddot\theta=-\kappa\sin(\theta-\varphi),\qquad I_2\ddot\varphi=\kappa\sin(\theta-\varphi).
$$
A static <equilibrium point> has both angular velocities zero and $\sin(\theta-\varphi)=0$. Hence the complete equilibrium families, with angles taken modulo $2\pi$, are
$$
\boxed{(\theta,\varphi)=(q,q)\quad\text{or}\quad(\theta,\varphi)=(q+\pi,q),\qquad q\in\mathbb R/(2\pi\mathbb Z).}
$$
The first family is aligned in azimuth and minimizes the spring length; the second is antipodal and maximizes it. Their potentials are $k[(a-A)^2+h^2]/2$ and $k[(a+A)^2+h^2]/2$, respectively. Common rotations generate a whole family rather than isolated equilibria.
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