  Recent experiments on strongly correlated bilayer quantum Hall systems
strongly suggest that, contrary to the usual assumption, the electron spin
degree of freedom is not completely frozen either in the quantum Hall or in the
compressibles states that occur at filling factor $\nu =1.$ These experiments
imply that the quasiparticles at $\nu =1$ could have both spin and pseudospin
textures i.e. they could be CP$^{3}$ skyrmions. Using a microscopic
unrestricted Hartree-Fock approximation, we compute the energy of several
crystal states with spin, pseudospin and mixed spin-pseudospin textures around
$\nu =1$ as a function of interlayer separation $d$ for different values of
tunneling ($\Delta_{SAS}$), Zeeman ($\Delta_{Z}$), and bias ($\Delta_{b}$)
energies. We show that in some range of these parameters, crystal states
involving a certain amount of spin depolarization have lower energy than the
fully spin polarized crystals. We study this depolarization dependence on
$d,\Delta_{SAS},\Delta_{Z}$ and $\Delta_{b}$ and discuss how it can lead to the
fast NMR relaxation rate observed experimentally.
