  A fibration is said to be isotrivial if all of its smooth fibres are
isomorphic to a single fixed variety. We classify the elliptic K3 surfaces that
are isotrivial, and use them to construct Lagrangian fibrations that are
isotrivial. We then modify the construction to produce new examples of
holomorphic symplectic orbifolds, that also admit isotrivial Lagrangian
fibrations.
