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

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