Poncelet pairs of a circle and parabolas from a confocal family and Painlevé VI equations

We study pairs of conics $(\mathcal{D},\mathcal{P})$, called $n\mathit{\text{-Poncelet}}\mathit{\text{pairs}}$, such that an $n$-gon, called an $n\mathit{\text{-Poncelet}}\mathit{\text{polygon}}$, can be inscribed into $\mathcal{D}$ and circumscribed about $\mathcal{P}$. Here $\mathcal{D}$ is a circle and $\mathcal{P}$ is a parabola from a confocal pencil $\mathcal{F}$ with the focus $F$. We prove that the circle contains $F$ if and only if every parabola $\mathcal{P}\in \mathcal{F}$ forms a $3$-Poncelet pair with the circle. We prove that the center of $\mathcal{D}$ coincides with $F$ if and only if every parabola $\mathcal{P}\in \mathcal{F}$ forms a $4$-Poncelet pair with the circle. We refer to such property, observed for $n=3$ and $n=4$, as $n\mathit{\text{-isoperiodicity}}$. We prove that $\mathcal{F}$ is not $n$-isoperiodic with any circle $\mathcal{D}$ for $n$ different from $3$ and $4$. Using isoperiodicity, we construct explicit algebraic solutions to Painlevé VI equations.