Real-analyticity conjecture for coefficients of two-dimensional superintegrable metrics

Let $M$ be a connected two-dimensional manifold with a $C^\infty$ metric

$$g=\lambda(x_1,x_2)(dx_1^2+dx_2^2),$$

and suppose it is superintegrable with polynomial-in-momenta integrals

$$A=a_0(x_1,x_2)p_1^n+a_1(x_1,x_2)p_1^{n-1}p_2+\cdots+a_n(x_1,x_2)p_2^n,$$ $$B=b_0(x_1,x_2)p_1^k+b_1(x_1,x_2)p_1^{k-1}p_2+\cdots+b_k(x_1,x_2)p_2^k.$$

Real-analyticity conjecture. On the complement of a discrete set of points, the functions $\lambda$, $a_i$, and $b_j$ are real-analytic in the variables $x_1,x_2$.

This is the paper’s stated ultimate goal and gives a more precise local form of the real-analyticity claim for superintegrable metrics. Its status is not resolved in the supplied text.