  Let $G$ be a reductive algebraic group over an algebraically closed field $k$
of prime characteristic not $2$, whose Lie algebra is denoted $\mathfrak{g}$.
We call a subvariety $\mathfrak{X}$ of the nilpotent cone $N \subset
\mathfrak{g}$ monogamous if for every $e\in \mathfrak{X}$, the
$\mathfrak{sl}_2$-triples $(e,h,f)$ with $f\in \mathfrak{X}$ are conjugate
under the centraliser $C_G(e)$. Building on work by the first two authors, we
show there is a unique maximal closed $G$-stable monogamous subvariety $V
\subset N$ and that it is an orbit closure, hence irreducible. We show that $V$
can also be characterised in terms of Serre's $G$-complete reducibility.
