  We seek to characterize homology classes of Lagrangian projective spaces
embedded in irreducible holomorphic-symplectic manifolds, up to the action of
the monodromy group. This paper addresses the case of manifolds
deformation-equivalent to the Hilbert scheme of length-three subschemes of a K3
surface. The class of the projective space in the cohomology ring has
prescribed intersection properties, which translate into Diophantine equations.
Possible homology classes correspond to integral points on an explicit elliptic
curve; our proof entails showing that the only such point is two-torsion.
