  After projection to the lowest Landau level translational invariance and
particle conservation combine into dipole symmetry. We show that the new
symmetry forbids spontaneous $U(1)$ symmetry breaking at zero temperature. In
the case of the spatially inhomogeneous magnetic field, where the translational
invariance is absent, we show that the dipole symmetry disappears and the
constraint on the symmetry breaking is lifted. We pay special attention to the
fate of the Girvin-Macdonald-Platzman algebra in the inhomogeneous magnetic
field and show that a natural generalization of it is still present even though
the dipole symmetry is not.
