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 ∈ { 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.