In a 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 nondegeneracy of these travelling waves, which is a key step in their classification and for the construction of multitravelling waves
