  We study the one parameter family of potential functions $q\varphi^u$
associated with the geometric potential $\varphi^u$ for the geodesic flow of a
compact rank 1 surface of nonpositive curvature. For $q<1$ it is known that
there is a unique equilibrium state associated with $q\varphi^u$, and it has
full support. For $q > 1$ it is known that an invariant measure is an
equilibrium state if and only if it is supported on the singular set. We study
the critical value $q=1$ and show that the ergodic equilibrium states are
either the restriction to the regular set of the Liouville measure, or measures
supported on the singular set. In particular, when~$q = 1$, there is a unique
ergodic equilibrium state that gives positive measure to the regular set.
