for any lie group $g$, we construct a $g$\--equivariant analogue of symplectic capacities and give examples when $g=\mathbb{t}^k\times\mathbb{r}^{d-k}$, in which case the capacity is an invariant of integrable systems. then we study the continuity of these capacities, using the natural topologies on the symplectic $g$\--categories on which they are defined. this work is joint with alvaro pelayo and alessio figalli.