Let $X$ be a metric space, $\mu$ a Borel probability measure, and $T:X\rightarrow X$ be an ergodic measure preserving isometry.

Is $(X,\mu,T)$ measure theoretically isomorphic to a minimal isometry on a compact metric space (equivalently it has discrete spectrum)?

I understand Krieger representation theorem states that ergodic MPT are measure theoretically isomorphic to minimal systems on a compact metric spaces. I would like to know if structure like isometry can be conserved.

The general question I am interested in is: do ergodic isometries have discrete spectrum?