The problem about periodic solutions for the
family of linear differential equation
$$
L u\equiv \left(\frac{\partial}{i\partial t} - a\Delta
\right)u(x,t)=\nu G(u-f)$$ is considered on the multidimensional
sphere $x \in S^n$ under the periodicity condition
$u|_{t=0}=u|_{t=b}$ and $\int_{S^n}u(x,t)dx=0.$
Here $a$ is given real, $\nu$ 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 ${\Bbb C}\times {\Bbb R}^+.$
