A refinement of the two-radius theorem on the Bessel–Kingman hypergroup

In the present paper, we study an equation of the form
$$ \int_{0}^{r}T^\alpha_yf(x)x^{2\alpha+1}\,dx=0, \qquad |y|< R-r, \quad 0<r<R, $$
where $\alpha>-1/2$, $T^\alpha_y$ is the generalized Bessel translation operator, and $f$ is an even function locally integrable with respect to the measure $|x|^{2\alpha+1}\,dx$ on the interval $(-R,R)$. A description of the solutions of this equation in the form of series in special functions is obtained. Based on this result, we completely study the existence of a nonzero solution of a system of two such equations.