A metric minimal flow whose enveloping semigroup contains finitely many minimal ideals is PI

We show that a minimal metric flowX whose enveloping semigroupE(X), contains finitely many minimal ideals, is a PI-flow; i.e.X has a proximal extensionX' which can be built by iterating proximal and isometric extensions (starting with the trivial one point flow). An example is given which shows that the converse theorem does not hold. Finally, we show that ifX is a minimal, non-trivial, metric, weakly mixing flow and the group action is nilpotent thenE(X) contains infinitely many minimal ideals.