  We study Lagrangian points on smooth holomorphic curves in T${\mathbb P}^1$
equipped with a natural neutral K\"ahler structure, and prove that they must
form real curves. By virtue of the identification of T${\mathbb P}^1$ with the
space ${\mathbb L}({\mathbb E}^3)$ of oriented affine lines in Euclidean
3-space ${\mathbb E}^3$, these Lagrangian curves give rise to ruled surfaces in
${\mathbb E}^3$, which we prove have zero Gauss curvature.
  Each ruled surface is shown to be the tangent lines to a curve in ${\mathbb
E}^3$, called the edge of regression of the ruled surface. We give an
alternative characterization of these curves as the points in ${\mathbb E}^3$
where the number of oriented lines in the complex curve $\Sigma$ that pass
through the point is less than the degree of $\Sigma$. We then apply these
results to the spectral curves of certain monopoles and construct the ruled
surfaces and edges of regression generated by the Lagrangian curves.
