  Let $(G/\Gamma,R_a)$ be an ergodic $k$-step nilsystem for $k\geq 2$. We adapt
an argument of Parry to show that $L^2(G/\Gamma)$ decomposes as a sum of a
subspace with discrete spectrum and a subspace of Lebesgue spectrum with
infinite multiplicity. In particular, we generalize a result previously
established by Host, Kra and Maass for $2$-step nilsystems and a result by
Stepin for nilsystems $G/\Gamma$ with connected, simply connected $G$.
