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
$$ (L-\lambda)u\equiv\biggl(\frac1i\frac\partial{\partial t}-a\Delta-\lambda\biggr)u(x,t)=\nu G(u-f) $$
is considered on the multidimensional sphere $x\in S^n$ under the periodicity condition $u|_{t=0}=u|_{t=b}$. Here $a$ and $\lambda$ are given reals, $\nu$ is a fixed complex number, $Gu(x,t)$ 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 $\mathbb C\times\mathbb R^+$.