  We study Poncelet polygons inscribed in a given boundary conic and tangent to
conics from a confocal pencil, when the boundary does not belong to the pencil.
This question naturally arose in the study of the numerical range and Blaschke
products. We compare and contrast the obtained results with the case when the
boundary belongs to the confocal pencil of caustics, which was studied before,
in particular in the study of elliptical billiards. We examine the monotonicity
of rotation numbers argument and observe deficiencies in typical ways such
arguments were used. We derive and use Cayley-type conditions to fill the gaps
in the existing proofs and derive more general statements. We discover confocal
families of conics with the property that each conic from the family is a
caustic for $k$-Poncelet polygons inscribed in the circle, with the same $k$.
For $k=4$ we characterize all such families. Then, we relate them to solutions
to Painlev\'e VI equations. We prove that such families do not exist for $k\ne
4$.
