2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/120590Bubbles of inviscid fluid surrounded by a viscous fluid in a Hele-Shaw cell can merge and break-off. During the process of break-off, a thinning neck pinches off to a universal self-similar singularity. We describe this process and reveal its integrable structure: it is a solution of the dispersionless limit of the AKNS hierarchy. The singular break-off patterns are universal, not sensitive to details of the process and can be seen experimentally. We briefly discuss the dispersive regularization of the Hele-Shaw problem and the emergence of the Painlevé II equation at the break-off.27 pages, 9 figures; typo correctedExactly Solvable and Integrable SystemsSoft Condensed MatterHigh Energy Physics - TheoryPattern Formation and Solitonstext