  The sigma-model of closed strings spinning in the $\eta$-deformation of
$AdS_{5} \times S^{5}$ leads to an integrable deformation of the
one-dimensional Neumann-Rosochatius mechanical system. In this article we
construct general solutions to this system that can be written in terms of
elliptic functions. The solutions correspond to closed strings with
non-constant radii rotating with two different angular momenta in an
$\eta$-deformed three-sphere. We analyse the reduction of the elliptic
solutions for some limiting values of the deformation parameter. For the case
of solutions with constant radii we find the dependence of the classical energy
of the string on the angular momenta as an expansion in the 't Hooft coupling.
