  We investigate structured grids aligned to the contours of a two-dimensional
flux-function with an X-point (saddle point). Our theoretical analysis finds
that orthogonal grids exist if and only if the Laplacian of the flux-function
vanishes at the X-point. In general, this condition is sufficient for the
existence of a structured aligned grid with an X-point. With the help of
streamline integration we then propose a numerical grid construction algorithm.
In a suitably chosen monitor metric the Laplacian of the flux-function vanishes
at the X-point such that a grid construction is possible.
  We study the convergence of the solution to elliptic equations on the
proposed grid. The diverging volume element and cell sizes at the X-point
reduce the convergence rate. As a consequence, the proposed grid should be used
with grid refinement around the X-point in practical applications. We show that
grid refinement in the cells neighboring the X-point restores the expected
convergence rate.
