  We classify equivariant $\mathbb{C}^*$-actions on moduli spaces of Higgs
bundles corresponding to the Painlev\'e equations. We deduce the
Floer-theoretic filtrations on the cohomology of these spaces, introduced by
Ritter and the second author in arXiv:2304.13026. Filtration is then compared
with the $``P=W"$ and the filtration obtained by multiplicities of the
irreducible components of the nilpotent cone, for all the 2-dimensional Higgs
moduli.
