  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, we
consider both the boundary rigidity problem and scattering rigidity problem,
see the introduction for details. We show that these two problems are
equivalent on simple $\MP$-systems. Unlike the cases of geodesic or magnetic
systems, knowing boundary action functions or scattering relations for only one
energy level is insufficient to uniquely determine a simple $\MP$-system, even
under 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 solved
up to an isometry and a gauge transformations of $\alpha$ and $U$. We prove
rigidity results for metrics in a given conformal class, for simple real
analytic $\MP$-systems and for simple two-dimensional $\MP$-systems.
