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We prove that almost every interval exchange transformation, with anassociated translation surface of genus $g\geq 2$, can be non-trivially andisometrically embedded in a family of piecewise isometries. In particular thisproves the existence of invariant curves for piecewise isometries, reminiscentof KAM curves for area preserving maps, which are not unions of circle arcs orline segments.

Existence of non-trivial embeddings of Interval Exchange Transformations into Piecewise Isometries