  We establish existence and uniqueness results for the modified binormal
curvature flow equation that generalizes the binormal curvature flow equation
for a curve in $\R^3.$ In this generalization, the velocity of the curve is
still directed along the binormal vector, but the magnitude of the speed is
allowed to depend on th parametrization of the curve, the time and the position
of the point in the space. We achieve our objective via a generalized form of
the Schr\"odinger map equation.
