*Preprint*

**Inserted:** 30 sep 2024

**Last Updated:** 30 sep 2024

**Year:** 2024

**Abstract:**

We consider entire solutions $\omega\in \dot{H}^1(\mathbb{R}^2;\mathbb{R}^3)$ of the $H$-system $\Delta\omega=2\omega_x\wedge\omega_y$, 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.