Capillary-gravity water waves with compactly supported vorticity II. Continuation and folds of the solution curve

This five-part series establishes the first systematic theory of capillary--gravity water waves carrying compactly supported vorticity with fixed positive circulation. Treating the free surface and the vortex as coupled unknowns, we study the well-posedness of traveling-wave equilibria, including the existence and continuation of solution branches with respect to bifurcation parameters, as well as their variational structure, geometric rigidity, canonical classification, and conditional dynamics. In this second paper of the series, we apply the Haziot-Wheeler analytic continuation theorem to extend the small-speed point-vortex branch constructed in Part I to a global curve of coupled equilibria. We establish two alternatives: either the geometric control quantity diverges, or it remains bounded along a sequence of solutions with no strongly convergent subsequence. If the second alternative occurs, we use concentration--compactness to obtain a finite profile decomposition without assuming monotonicity of the free surface: a nontrivial centered vortex wave and finitely many reflected pairs of irrotational waves, with a remainder converging strongly to zero. At any simple speed fold, a Schur-complement reduction in the physical vortex coordinate shows that the static energy-momentum Hessians of the two nearby equal-speed waves differ by one in negative index within the symmetric class. Finally, we use a joint two-parameter Lyapunov--Schmidt reduction to extend the regularization of Part I to compact arcs of the global solution curve, carrying the canonical smooth-core branches through their specified simple speed folds. These results provide a global point-vortex reference for the variational construction in Part III and canonical smooth-core comparison arcs for the rigidity and uniqueness theory in Part IV.