On quadrilateral orbits in planar billiards

V. Ya. Ivrii's conjecture (1980) states that in every piecewice-smooth billiard in euclidean space the set of periodic orbits has measure zero. This conjecture is closely related to G. Weyl's conjecture from the spectral theory of the laplacian. Particular cases of Ivrii's conjecture for triangular orbits were proved by many authors, first of all, M. Rychlik (1989, in dimension two) and Ya. B. Vorobets (1991, in every dimension). The particular case for quadrilateral orbits in dimension two was proved in a joint work of Yu. G. Kudryashov and the speaker (for piecewise-smooth billiards with pieces smooth enough).
We will discuss partial results on the two-dimensional complex-algebraic version of the Ivrii's conjecture, for reflections with respect to complex planar algebraic curves. It appears that in this context even the Ivrii's conjecture for quadrilateral orbits is not true, and the counterexamples can be described. The nontrivial counterexamples are the billiards formed by pairs of confocal conics.