  We study the geodesic flow on the normal line congruence of a minimal surface
in ${\Bbb{R}}^3$ induced by the neutral K\"ahler metric on the space of
oriented lines. The metric is lorentz with isolated degenerate points and the
flow is shown to be completely integrable. In addition, we give a new
holomorphic description of minimal surfaces in ${\Bbb{R}}^3$ and relate it to
the classical Weierstrass representation.
