  The famous conjecture of V.Ya.Ivrii says that {\it in every billiard with
infinitely-smooth boundary in a Euclidean space the set of periodic orbits has
measure zero}. In the present paper we study its complex analytic version for
quadrilateral orbits in two dimensions, with reflections from holomorphic
curves. We present the complete classification of 4-reflective complex analytic
counterexamples: billiards formed by four holomorphic curves in the projective
plane that have open set of quadrilateral orbits. This extends the previous
author's result classifying 4-reflective complex planar algebraic
counterexamples. We provide applications to real planar billiards:
classification of 4-reflective germs of real planar $C^4$-smooth
pseudo-billiards; solutions of Tabachnikov's Commuting Billiard Conjecture and
the 4-reflective case of Plakhov's Invisibility Conjecture (both in two
dimensions; the boundary is required to be piecewise $C^4$-smooth). We provide
a survey and a small technical result concerning higher number of complex
reflections.
