  Given two compact manifolds $X,Y,$ with boundary and a boundary preserving
symplectomorphism $\chi:T^*Y\setminus0\to T^*X\setminus0$, which is
one-homogeneous in the fibers and satisfies the transmission condition, we
introduce Fourier integral operators of Boutet de Monvel type associated with
$\chi$. We study their mapping properties between Sobolev spaces, develop a
calculus and prove a Egorov type theorem. We also introduce a notion of
ellipticity which implies the Fredholm property. Finally, we show how -- in the
spirit of a classical construction by A. Weinstein -- a Fredholm operator of
this type can be associated with $\chi$ and a section of the Maslov bundle. If
$\dim Y>2$ or the Maslov bundle is trivial, the index is independent of the
section and thus an invariant of the symplectomorphism.
