  In this article, we give a survey of Geometric Invariant Theory for Toric
Varieties, and present an application to the Einstein-Weyl Geometry. We compute
the image of the Minitwistor space of the Honda metrics as a categorical
quotient according to the most efficient linearization. The result is the
complex weighted projective space CP_(1,1,2). We also find and classify all
possible quotients.
