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In this paper, we consider a compact Riemannian manifold with boundary,endowed with a magnetic potential $\alpha$ and a potential $U$. For brevity,this type of systems are called $\MP$-systems. On simple $\MP$-systems, weconsider both the boundary rigidity problem and scattering rigidity problem,see the introduction for details. We show that these two problems areequivalent on simple $\MP$-systems. Unlike the cases of geodesic or magneticsystems, knowing boundary action functions or scattering relations for only oneenergy level is insufficient to uniquely determine a simple $\MP$-system, evenunder the assumption that we know the restriction of the system on the boundary$\p M$, and we provide some counterexamples. These problems can only be solvedup to an isometry and a gauge transformations of $\alpha$ and $U$. We proverigidity results for metrics in a given conformal class, for simple realanalytic $\MP$-systems and for simple two-dimensional $\MP$-systems.

Boundary and scattering rigidity problems in the presence of a magnetic field and a potential