  The Cauchy problem for harmonic maps from Minkowski space with its standard
flat metric to a certain non-constant curvature Lorentzian 2-metric is studied.
The target manifold is distinguished by the fact that the Euler-Lagrange
equation for the energy functional is Darboux integrable. The time evolution of
the Cauchy data is reduced to an ordinary differential equation of Lie type
associated to SL(2) acting on a manifold of dimension 4. This is further
reduced to the simplest Lie system: the Riccati equation. Lie reduction permits
explicit representation formulas for various initial value problems.
Additionally, a concise (hyperbolic) Weierstrass-type representation formula is
derived. Finally, a number of open problems are framed.
