Question

What does it mean for a system to be integrable?

Answer 1

The paper refers to the notion of complete integrability, aka. Liouville integrability, i.e. the property that there exist nn independent globally defined Poisson-commuting constants of motion. Here 2n2n is the dimension of phase space. In the paper n=2n=2. See also this related Phys.SE post.

Answer 2

Dedecker's paper "Intégrales complètes de l'équation aux dérivées partielles de Hamilton-Jacobi d'une intégrale multiple", C.R. Acad. Sc. Paris, 285 (1977) pp. 123-6. Together with two preceding notes in the same journal, this generalises the concept of "complete integrability" of a mechanical system.