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

V. Dragović^ab, M. H. Murad<sup>ac

a</sup> Department of Mathematical Sciences, The University of Texas at Dallas, Richardson, TX, USA
^b Mathematical Institute, Serbian Academy of Sciences and Arts, Belgrade, Serbia
^c Department of Mathematics and Natural Sciences, BRAC University, Dhaka, Bangladesh

Abstract: We study pairs of conics $(\mathcal{D},\mathcal{P})$, called $n$-Poncelet pairs, such that an $n$-gon, called an $n$-Poncelet 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 $n$-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.

Keywords: cyclic $n$-gons, confocal parabolas, $n$-Poncelet pairs, Cayley conditions, isorotational families, Painlevé VI equations.

MSC: Primary 14H70, 34M55; Secondary 37J70, 37A10, 51N20

Received: 20.01.2025
Revised: 18.03.2025

Language: English

DOI: 10.4213/im9697

 English version: Izvestiya: Mathematics, 2026, 90:1, 144–168