  We consider the effect of interactions on electrons confined to two
dimensions at Landau level filling $\nu=2$, with the specific aim to determine
the range of parameters where the fully polarized state is stable. We calculate
the charge and the spin density collective modes in random phase approximation
(RPA) including vertex corrections (also known as time dependent Hartree Fock),
and treating the Landau level mixing accurately within the subspace of a single
particle hole pair. It is found that the spin wave excitation mode of the fully
polarized state has a roton minimum which deepens as a result of the
interaction induced Landau level mixing, and the energy of the roton vanishes
at a critical Zeeman energy signaling an instability of the fully polarized
state at still lower Zeeman energies. The feasibility of the experimental
observation of the roton minimum in the spin wave mode and its softening will
be discussed. The spin and charge density collective modes of the unpolarized
state are also considered, and a phase diagram for the $\nu =2$ state as a
function of $r_{S}$ and the Zeeman energy is obtained.
