  We investigate the properties of the zeta-function of well-rounded
sublattices of a fixed arithmetic lattice in the plane. In particular, we show
that this function has abscissa of convergence at $s=1$ with a real pole of
order 2, improving upon a recent result of S. Kuehnlein. We use this result to
show that the number of well-rounded sublattices of a planar arithmetic lattice
of index less or equal $N$ is $O(N \log N)$ as $N \to \infty$. To obtain these
results, we produce a description of integral well-rounded sublattices of a
fixed planar integral well-rounded lattice and investigate convergence
properties of a zeta-function of similarity classes of such lattices, building
on some previous results of the author.
