On the structure of the set of periods for periodic solutions of some linear integro-differential equations on the multidimensional sphere

The problem of periodic solutions for the family of linear differential equations

$(L-{\lambda})u\equiv \Big(\frac{1}{i}\frac{\partial}{\partial t} - a\Delta- \lambda\Big) u(x,t)=\nu G(u-f)$

is considered on the multidimensional sphere $x\in S^n$ under the periodicity condition $u\vert _{t=0}=u\vert _{t=b}$. Here $a$ and $\lambda$ are given reals, $\nu$ is a fixed complex number, $G u(x,t)$ is a linear integral operator, and $\Delta$ is the Laplace operator on $S^n$. It is shown that the set of parameters $(\nu, b)$ for which the above problem admits a unique solution is a measurable set of full measure in ${\mathbb{C}} \times {\mathbb{R}}^+$.