  Let $X$ be a connected Riemannian manifold such that the resolvent of the
free Laplacian $(\Delta-z)^{-1}, z\in\C\setminus\R^{+},$ has a meromorphic
continuation through $\R^{+}$. The poles of this continuation are called
resonances. When $X$ has some symmetries, we construct complex-valued
potentials, $V$, such that the resolvent of $\Delta+V$, which has also a
meromorphic continuation, has the same resonances with multiplicities as the
free Laplacian.
