  We define minimal fusion systems in a way that every non-solvable fusion
system has a section which is minimal. Minimal fusion systems can also be seen
as analogs of Thompson's N-groups. In this paper, we consider a minimal fusion
system $\mathcal{F}$ on a finite $p$-group $S$ that has a unique maximal
$p$-local subsystem containing $N_{\mathcal{F}}(S)$. For an arbitrary prime
$p$, we determine the structure of a certain (explicitly described) $p$-local
subsystem of $\mathcal{F}$. If $p=2$, this leads to a complete classification
of the fusion system $\mathcal{F}$.
