  We extend the work of Schwarz [1] to show that bound states of type IIB
supersymmetric ($p, q$)-strings on a circle are associated with M2-branes
irreducibly wrapped on $T^2$, or equivalently with nontrivial worldvolume
fluxes. Beyond this extension we consider the Hamiltonian of an M2-brane with
$C_{\pm}$ fluxes formulated on a symplectic torus bundle with monodromy. In
particular, we analyze the relevant case when the monodromy is parabolic. We
show that the Hamiltonian is defined in terms of the coinvariant module. We
also find that the mass operator is invariant under transformations between
inequivalent coinvariants. These coinvariants classify the inequivalent classes
of twisted torus bundles with nontrivial monodromy for a given flux. We obtain
their associated ($p,q$)-strings via double dimensional reduction, which are
invariant under a parabolic subgroup of $SL(2,\mathbb{Q})$. This is the origin
of the gauge symmetry of the associated gauged supergravity. These bound states
could also be related to the parabolic Scherk-Schwarz reductions of type IIB
string theory.
