  Third rank Killing tensors in (1+1)-dimensional geometries are investigated
and classified. It is found that a necessary and sufficient condition for such
a geometry to admit a third rank Killing tensor can always be formulated as a
quadratic PDE, of order three or lower, in a Kahler type potential for the
metric. This is in contrast to the case of first and second rank Killing
tensors for which the integrability condition is a linear PDE. The motivation
for studying higher rank Killing tensors in (1+1)-geometries, is the fact that
exact solutions of the Einstein equations are often associated with a first or
second rank Killing tensor symmetry in the geodesic flow formulation of the
dynamics. This is in particular true for the many models of interest for which
this formulation is (1+1)-dimensional, where just one additional constant of
motion suffices for complete integrability. We show that new exact solutions
can be found by classifying geometries admitting higher rank Killing tensors.
