  Reflection in strictly convex bounded planar billiard acts on the space of
oriented lines and preserves a standard area form. A caustic is a curve $C$
whose tangent lines are reflected by the billiard to lines tangent to $C$. The
famous Birkhoff conjecture states that the only strictly convex billiards with
a foliation by closed caustics near the boundary are ellipses. By Lazutkin's
theorem, there always exists a Cantor family of closed caustics approaching the
boundary. In the present paper we deal with billiards whose boundary contains a
strictly convex embedded (non-closed) curve $\gamma$. We prove that there
exists a domain adjacent to $\gamma$ from the convex side that is
$C^{\infty}$-smoothly foliated by non-closed caustics. This generalizes a
previous result by R.Melrose, which yields existence of a local foliation by
caustics in a neighborhood of a boundary point. We show that there exists a
continuum of above foliations by caustics whose germs at each point in $\gamma$
are pairwise different. We prove a more general version of this statement in
the case, when both $\gamma$ and the caustics are immersed curves. It also
applies to a billiard bounded by a closed strictly convex curve $\gamma$ and
yields infinitely many "immersed" foliations by immersed caustics. For the
proof of the above results, we state and prove their analogue for a special
class of area-preserving maps generalizing billiard reflections: the so-called
$C^{\infty}$-lifted strongly billiard-like maps.
