  We prove that almost every interval exchange transformation, with an
associated translation surface of genus $g\geq 2$, can be non-trivially and
isometrically embedded in a family of piecewise isometries. In particular this
proves the existence of invariant curves for piecewise isometries, reminiscent
of KAM curves for area preserving maps, which are not unions of circle arcs or
line segments.
