  For any minimal system $(X,T)$ and $d\geq 1$ there is an associated minimal
system $(N_{d}(X), \mathcal{G}_{d}(T))$, where $\mathcal{G}_{d}(T)$ is the
group generated by $T\times\cdots\times T$ and $T\times T^2\times\cdots\times
T^{d}$ and $N_{d}(X)$ is the orbit closure of the diagonal under
$\mathcal{G}_{d}(T)$. It is known that the maximal $d$-step pro-nilfactor of
$N_d(X)$ is $N_d(X_d)$, where $X_d$ is the maximal $d$-step pro-nilfactor of
$X$.
  In this paper, we further study the structure of $N_d(X)$. We show that the
maximal distal factor of $N_d(X)$ is $N_d(X_{dis})$ with $X_{dis}$ being the
maximal distal factor of $X$, and prove that as minimal systems $(N_{d}(X),
\mathcal{G}_{d}(T))$ has the same structure theorem as $(X,T)$. In addition, a
non-saturated metric example $(X,T)$ is constructed, which is not $T\times
T^2$-saturated and is a Toeplitz minimal system.
