A translation surface is a closed orientable surface equipped, in the complement of a finite set of singularities, with an atlas whose transition maps are translation of \(\mathbb{R}^2\). Such a surface inherits a flat structure with conical singularities and it is naturally equipped with two orthogonal singular foliations.
In each direction (with respect to the angle structure defined by these two foliations), there is a flow on the surface, defined in the complement of the singular leaves. Each such flow determines a decomposition of the surface into invariant sets on which the restricted flow is either periodic or minimal.
For each fixed genus, there is a moduli space of translation surfaces which is equipped with a natural stratification, where the strata are parametrized by the singularities. In the paper under review, the author studies this decomposition for translation surfaces in the hyperelliptic connected components of the moduli space \(\mathcal{H}^{hyp}(2g-2)\) and \(\mathcal{H}^{hyp}(g-1,g-1)\) of the corresponding strata. Specifically, she obtains a characterization of the pairs of nonnegative integers \((p,m)\) for which there exists a translation surface in \(\mathcal{H}^{hyp}(2g-2)\) and \(\mathcal{H}^{hyp}(g-1,g-1)\) with precisely \(p\) periodic components and \(m\) minimal components.