  We show the relation between three non trivial sectors of M2-brane theory
formulated in the LCG connected among them by canonical transformations. These
sectors correspond to the supermembrane theory formulated on a $M_9\times T^2$
on three different constant three-form backgrounds: M2-brane with constant
$C_{-}$, M2-brane with constant $C_{\pm}$ and M2-brane with a generic constant
$C_3$ denoted as CM2-brane. The first two exhibit a purely discrete
supersymmetric spectrum once the central charge condition, or equivalently, the
corresponding flux condition has been turned on. The CM2-brane is conjectured
to share this spectral property once that fluxes $C_{\pm}$ are turned on. As
shown in [1] they are duals to three inequivalent sectors of the D2-branes with
specific worldvolume and background RR and NSNS quantization conditions on each
case.
