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We classify equivariant 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.
Szabó et al. (Thu,) studied this question.