  In the previous paper, we constructed a smooth branch of travelling waves for
the 2 dimensional Gross-Pitaevskii equation. Here, we continue the study of
this branch. We show some coercivity results, and we deduce from them the
kernel of the linearized operator, a spectral stability result, as well as a
uniqueness result in the energy space. In particular, our result proves the non
degeneracy of these travelling waves, which is a key step in the classification
of these waves and for the construction of multi-travelling waves.
