  A theory of single-element extensions of integer polymatroids analogous to
that of matroids is developed. We present an algorithm to generate a catalog of
$2$-polymatroids, up to isomorphism. When we implemented this algorithm on a
computer, obtaining all $2$-polymatroids on at most seven elements, we
discovered the surprising fact that the number of $2$-polymatroids on seven
elements fails to be unimodal in rank.
