On the existence of degenerate solutions of the two-dimensional $H$-system

We consider entire solutions $\omega \in {\dot{H}}^1({\mathbb{R}}^2;{\mathbb{R}}^3)$ of the $H$-system $\mathrm{\Delta }\omega =2{\omega }_x\wedge {\omega }_y,$ which we refer to as bubbles. Surprisingly, and contrary to conjectures raised in the literature, we find that bubbles with degree at least three can be degenerate: the linearized $H$-system around a bubble can admit solutions that are not tangent to the smooth family of bubbles. We then give a complete algebraic characterization of degenerate bubbles.