A Lagrangian fibration of a projective hyper-Kähler manifold is called isotrivial if its smooth (abelian variety) fibers are all isomorphic to each other. Given an isotrivial fibration of a HK manifold f : X -> B with at least one rational section, we prove the following four descriptions of the fibration: (1) The common abelian variety fiber F is isogenous to the power of an elliptic curve E. (2) There are two types (called type A and B) of isotrivial fibrations with different behaviors. (3) If the elliptic curve factor E of the fiber has no CM by Q(\sqrt{-1}) or Q(\sqrt{-3}), then the fibration is necessarily of type A. (4) Every type A isotrivial fibration is birational to one of the two explicit constructions of isotrivial HK, starting from an elliptic K3 surface or an abelian surface. Our proof of the last statement (4) depends on the regularity conjecture on the base of a Lagrangian fibration. This is joint work with Radu Laza and Olivier Martin.