  We investigate distribution of integral well-rounded lattices in the plane,
parameterizing the set of their similarity classes by solutions of the family
of Pell-type Diophantine equations of the form $x^2+Dy^2=z^2$ where $D>0$ is
squarefree. We apply this parameterization to the study of the greatest minimal
norm and the highest signal-to-noise ratio on the set of such lattices with
fixed determinant, also estimating cardinality of these sets (up to rotation
and reflection) for each determinant value. This investigation extends previous
work of the first author in the specific cases of integer and hexagonal
lattices and is motivated by the importance of integral well-rounded lattices
for discrete optimization problems. We briefly discuss an application of our
results to planar lattice transmitter networks.
