  In this work we obtain classical solutions of the bosonic sector of the
supermembrane theory with two-form fluxes associated to a quantized constant
$C_{\pm}$ background. This theory satisfies a flux condition on the worldvolume
that induces monopoles over it. Classically it is stable as it does not contain
string-like spikes with zero energy in distinction with the general case. At
quantum level the bosonic membrane has a purely discrete spectrum but the
relevance is that the same property holds for its supersymmetric spectrum. We
find for this theory spinning membrane solutions, some of them including the
presence of a non-vanishing symplectic gauge connection defined on its
worldvolume in different approximations. By using the duality found between
this theory and the so-called supermembrane with central charges, rotating
membrane solutions found in that case, are also solutions of the M2-brane with
$C_{\pm}$ fluxes. We generalize this result to other embeddings. We find new
distinctive rotating membrane solutions, some of them including the presence of
a non-vanishing symplectic gauge connection defined on its worldvolume. We
obtain numerical and analytical solutions in different approximations
characterizing the dynamics of the membrane with fluxes $C_{\pm}$ for different
ans\"atze of the dynamical degrees of freedom. Finally we discuss the physical
admissibility of some of these ans\"atze to model the components of the
symplectic gauge field.
