  The combined effect of a lateral square superlattice potential and the
Coulomb interaction on the ground state of a two-dimensional electron gas in a
perpendicular magnetic field is studied for different rational values of
$\Gamma$, the inverse of the number of flux quanta per unit cell of the
external potential, at filling factor $\nu =1$ in Landau level $N=0.$ When
Landau level mixing and disorder effects are neglected, increasing the strength
$W_{0}$ of the potential induces a transition, at a critical strength
$W_{0}^{\left( c\right) },$ from a uniform and fully spin polarized state to a
two-dimensional charge density wave (CDW) with a meronlike spin texture at each
maximum and minimum of the CDW. The collective excitations of this vortex-CDW
are similar to those of the Skyrme crystal that is expected to be the ground
state near filling factor $\nu =1$. In particular, a broken U(1) symmetry in
the vortex-CDW results in an extra gapless phase mode that could provide a fast
channel for the relaxation of nuclear spins. The average spin polarization $%
S_{z}$ changes in a continuous or discontinuous manner as $W_{0}$ is increased
depending on whether $\Gamma \in \left[ 1/2,1\right] $ or $\Gamma \in \left[
0,1/2\right] .$ The phase mode and the meronlike spin texture disappear at
large value of $W_{0},$ leaving as the ground state a partially spin-polarized
CDW if $\Gamma \neq 1/2$ or a spin-unpolarized CDW if $\Gamma =1/2.$
