  Lagrangian fibrations of hyperk\"ahler manifolds are induced by semi-ample
line bundles which are isotropic with respect to the Beauville-Bogomolov-Fujiki
form. For a non-isotrivial family of hyperk\"ahler manifolds over a complex
manifold $S$ of positive dimension, we prove that the set of points in $S$, for
which there is an isotropic class in the Picard lattice of the corresponding
hyperk\"ahler manifold represented as a fiber over that point, is analytically
dense in $S$. We also prove the expected openness and density of the locus of
polarised hyperk\"ahler manifolds that admit a nef algebraic isotropic line
bundle.
