Isoperiodic families of Poncelet polygons inscribed in a circle and circumscribed about conics from a confocal pencil

Poncelet polygons inscribed in a circle and circumscribed about conics from a confocal family naturally arise in the analysis of the numerical range and Blaschke products. We examine the behaviour of such polygons when the inscribed conic varies through a confocal pencil and discover cases when each conic from the confocal family is inscribed in an n-polygon, which is inscribed in the circle, with the same n. Complete geometric characterization of such cases for \(n\in \{4,6\}\) is given and proved that this cannot happen for other values of n. We establish a relationship of such families of Poncelet quadrangles and hexagons to solutions of a Painlevé VI equation.