  We give examples of compact symplectic manifolds with disconnected contact
type boundary in dimension $4n$ for any $n\geq 1$. The example is given by a
subset of the tangent bundle of a compact quotient of the complex hyperbolic
space endowed with the canonical symplectic form plus a generalized magnetic
field and its boundary is given by two hypersurfaces of constant kinetic
energy. In particular, when $n=1$, it is obtained by the tangent bundle of a
surface of genus greater than 1 endowed with the canonical symplectic form plus
the pullback of the K\"ahler form by the canonical projection.
