Ergodic components of an extension by a nilmanifold (Q733365)

Let \((T,X,\mu)\) be an ergodic dynamical system. It is shown that all ergodic components of an extension of this system by translations on a nilmanifold \(M\) are isomorphic to extensions of this system by translations on submanifolds of \(M\). It is worthwhile noting that this result is connected to a lemma by W. Parry which claims that a shift-transformation of a compact connected nilmanifold \(M\) is ergodic if and only if it is ergodic on the maximal factor-torus of~\(M\).