  This article initiates the study of isotrivial Lagrangian fibrations of
compact hyper-K\"ahler manifolds. We present four foundational results that
extend well-known facts about isotrivial elliptic fibrations of K3 surfaces.
First, we prove that smooth fibers of an isotrivial Lagrangian fibration are
isogenous to a power of an elliptic curve. Second, we exhibit a dichotomy
between two types of isotrivial Lagrangian fibrations, which we call A and B.
Third, we give a classification result for type A isotrivial Lagrangian
fibrations assuming a well-known conjecture on the base of Lagrangian
fibrations. Namely, if a type A isotrivial Lagrangian fibration admits a
rational section, then it is birational to one of two straightforward examples
of isotrivial fibrations of hyper-K\"ahler manifolds of $\text{K3}^{[n]}$-type
and $\text{Kum}_n$-type. Finally, we prove that a genericity assumption on the
smooth fiber of an isotrivial Lagrangian fibration ensures that the fibration
is of type A.
