  We study in detail entanglement properties of the Jaynes-Cummings model
assuming a two-level atom (qubit) interacting with the first $N$ levels of an
electromagnetic field mode (qudit) in a cavity. In the Jaynes-Cummings model,
the number operator is the conserved quantity that allows for the exact
diagonalization of the Hamiltonian and thus we study states that commute with
this conserved quantity and whose structure is preserved under the
Jaynes-Cummings dynamics. Contrary to the common belief, we show that there are
bound entangled states that satisfy the symmetries imposed by the conservation
of the number of excitations when $N>3$. Furthermore we show that \emph{the
Jaynes-Cummings interaction can be used to generate bound-entanglement} between
the atom and the mode.
