  In the generalized Legendre approach, the equation describing an
asymptotically locally Euclidean space of type $D_n$ is found to admit an
algebraic formulation in terms of the group law on a Weierstrass cubic. This
curve has the structure of a Cayley cubic for a pencil generated by two
transversal plane conics, that is, it takes the form $Y^2 = \det ({\cal
A}+X{\cal B})$, where ${\cal A}$ and ${\cal B}$ are the defining $3 \times 3$
matrices of the conics. In this light, the equation can be interpreted as the
closure condition for an elliptic billiard trajectory tangent to the conic
${\cal B}$ and bouncing into various conics of the pencil determined by the
positions of the monopoles. Poncelet's porism guarantees then that once a
trajectory closes to a star polygon, any trajectory will close, regardless of
the starting point and after the same number of steps.
