  Let $M$ be a hyperkaehler manifold, and $\eta$ a closed, positive (1,1)-form
which is degenerate everywhere on $M$. We associate to $\eta$ a family of
complex structures on $M$, called a degenerate twistor family, and parametrized
by a complex line. When $\eta$ is a pullback of a Kaehler form under a
Lagrangian fibration $L$, all the fibers of degenerate twistor family also
admit a Lagrangian fibration, with the fibers isomorphic to that of $L$.
Degenerate twistor families can be obtained by taking limits of twistor
families, as one of the Kahler forms in the hyperkahler triple goes to $\eta$.
