  Let $G$ be a connected reductive algebraic group defined over an
algebraically closed field $\mathbbm k$ of characteristic zero. We consider the
commuting variety $\mathcal C(\mathfrak u)$ of the nilradical $\mathfrak u$ of
the Lie algebra $\mathfrak b$ of a Borel subgroup $B$ of $G$. In case $B$ acts
on $\mathfrak u$ with only a finite number of orbits, we verify that $\mathcal
C(\mathfrak u)$ is equidimensional and that the irreducible components are in
correspondence with the {\em distinguished} $B$-orbits in $\mathfrak u$. We
observe that in general $\mathcal C(\mathfrak u)$ is not equidimensional, and
determine the irreducible components of $\mathcal C(\mathfrak u)$ in the
minimal cases where there are infinitely many $B$-orbits in $\mathfrak u$.
