Holomorphic solutions of the double confluent Heun equation associated with the RSJ model of the Josephson junction

This work is a continuation of research on a first-order nonlinear differential equation applied in the overshunted model of the Josephson junction. The approach is based on the relation between this equation and the double confluent Heun equation, which is a second-order linear homogeneous equation with two irregular singular points. We describe the conditions on the equation parameters under which its general solution is an analytic function on the Riemann sphere except at $0$ and $\infty$. We construct an explicit basis of the solution space. One of the functions in this basis is regular everywhere except $0$, and the other is regular everywhere except $\infty$. We show that in the framework of the RSJ model of Josephson junction dynamics, the described situation corresponds to the condition that the Shapiro step vanishes if all the solutions of the double confluent Heun equation are single-valued on the Riemann sphere without $0$ and $\infty$.