  We study the stationary points of what is known as the lattice Landau gauge
fixing functional in one-dimensional compact U(1) lattice gauge theory, or as
the Hamiltonian of the one-dimensional random phase XY model in statistical
physics. An analytic solution of all stationary points is derived for lattices
with an odd number of lattice sites and periodic boundary conditions. In the
context of lattice gauge theory, these stationary points and their indices are
used to compute the gauge fixing partition function, making reference in
particular to the Neuberger problem. Interpreted as stationary points of the
one-dimensional XY Hamiltonian, the solutions and their Hessian determinants
allow us to evaluate a criterion which makes predictions on the existence of
phase transitions and the corresponding critical energies in the thermodynamic
limit.
