  This note is a step towards demonstrating the benefits of a symplectic
approach to studying equivariant K\"ahler geometry. We apply a local
differential geometric framework from K\"ahler toric geometry due to Guillemin
& Abreu to the case of the standard linear $\SU(n)$ action on
$\bbC^n\setminus\{0\}$. Using this framework we (re)construct a scalar-flat
K\"ahler metric on the blow-up of $\bbC^n$ at the origin from data on the
moment polytope.
