  There exists a widely believed opinion, that the many-body ground state of a
two-dimensional electron system at a completely filled lowest Landau level (the
filling factor $\nu=1$) is described by the so-called Hartree-Fock wave
function, and that this solution is the unique, exact eigenstate of the system
at $\nu=1$. I show that this opinion is erroneous, construct an infinite number
of other variational many-body wave functions, and discuss the properties of a
few states which have the energy substantially lower than the energy of the
Hartree-Fock state.
