There are indications that in the 80s Guillemin posed a question: If billiard maps are conjugate, can we say that domains are the same up to isometry? On one side, we show that conjugacy of different domains can't be C^1 near the boundary. In particular, billiard maps of the circle and an ellipse are both analytically integrable, but not C^1 conjugate. On the other side, if conjugate near the boundary s smoother, then domains are the same up to isometry. (This is joint work with A. Sorrentino.)