Mutual isomorphisms of translations of a homogeneous flow

Ergodic one-parameter flows $(G/\Gamma,g_{\mathbb R})$ induced by the left action of a subgroup $g_{\mathbb R}\subset G$ on homogeneous spaces of finite volume are considered. Let $\mathscr M\subset{\mathbb R}^+$ be the set of all $t>0$ such that the cascade $(G/\Gamma,g_{t{\mathbb Z}})$ is metrically isomorphic to the cascade $(G/\Gamma,g_{\mathbb Z})$. We prove that either $\mathscr M$ is at most countable or the subgroup $g_\mathscr M$ is horocyclic and $\mathscr M={\mathbb R}^+$. We prove that a metric isomorphism of ergodic quasi-unipotent cascades (or flows) is affine on almost all fibers of a certain natural bundle. The result generalizes Witte's theorem on the affinity of such isomorphisms of cascades with the mixing property; this is applied to the study of the structure of the set $\mathscr M\subset{\mathbb R}^+$. The proof is based on the fundamental Ratner theorem stating that the ergodic measures of unipotent cascades are algebraic.