  We first improve an old result of McMahon and show that a metric minimal flow
whose enveloping semigroup contains less than $2^{\mathfrak{c}}$ (where
${\mathfrak{c}} ={2^{\aleph_0}}$) minimal left ideals is PI. Then we show the
existence of various minimal PI flows with many minimal left ideals, as
follows. For the acting group $G=SL_2(\mathbb{R})^\mathbb{N}$, we construct a
metric minimal PI $G$-flow with $\mathfrak{c}$ minimal left ideals. We then use
this example and results established in \cite{GW-79} to construct a metric
minimal PI cascade $(X,T)$ with $\mathfrak{c}$ minimal left ideals. We go on
and construct an example of a minimal PI-flow $(Y, \mathcal{G})$ on a compact
manifold $Y$ and a suitable path-wise connected group $\mathcal{G}$ of
homeomorphism of $Y$, such that the flow $(Y, \mathcal{G})$ is PI and has
$2^{\mathfrak{c}}$ minimal left ideals. Finally, we use this latter example and
a theorem of Dirb\'{a}k to construct a cascade $(X, T)$ which is PI (of order
3) and has $2^\mathfrak{c}$ minimal left ideals. Thus this final result shows
that, even for cascades, the converse of the implication "less than
$2^\mathfrak{c}$ minimal left ideals implies PI", fails.
