Asymptotics of solutions to the Sturm–Liouville equation along an arbitrary curve in a neighborhood of a symmetric singular point

Let $G$ be a convex domain with exactly one singular point $z_s$ of an analytic function $Q(z)$, symmetric with respect to this point. For large values of the modulus of the spectral parameter $\rho$, we study the asymptotics of the transfer matrix $P$ of the Sturm–Liouville equation with potential $Q$ along an arbitrary curve lying in the domain $G$ and not passing through the point $z_s$. Necessary and sufficient conditions are given under which the matrix $P$ is independent of the parameter $\rho$, and its structure is described in this case. In all other cases, we prove that every entry of the transfer matrix is an entire function of $\rho$ of completely regular growth of order $1/2$, with the same piecewise trigonometric indicator and angular density of zeros. Formulas for these characteristics are obtained for three possible types.