  In this paper we prove the Birkhoff-Poritsky conjecture for
centrally-symmetric $C^2$-smooth convex planar billiards. We assume that the
domain $\mathcal A$ between the invariant curve of $4$-periodic orbits and the
boundary of the phase cylinder is foliated by $C^0$-invariant curves. Under
this assumption we prove that the billiard curve is an ellipse. For the
original Birkhoff-Poritsky formulation we show that if a neighborhood of the
boundary of billiard domain has a $C^1$-smooth foliation by convex caustics of
rotation numbers in the interval (0; 1/4] then the boundary curve is an
ellipse. In the language of first integrals one can assert that {if the
billiard inside a centrally-symmetric $C^2$-smooth convex curve $\gamma$ admits
a $C^1$-smooth first integral with non-vanishing gradient on $\mathcal A$, then
the curve $\gamma$ is an ellipse.} The main ingredients of the proof are : (1)
the non-standard generating function for convex billiards discovered in
\cite{BM}, \cite{B}; (2) the remarkable structure of the invariant curve
consisting of $4$-periodic orbits; and (3) the integral-geometry approach
initiated in B0, B1 for rigidity results of circular billiards. Surprisingly,
we establish a Hopf-type rigidity for billiard in ellipse.
