  In this article we consider the Lagrangian mean curvature flow of compact,
circle-invariant, almost calibrated Lagrangian surfaces in hyperk\"ahler
4-manifolds with circle symmetry. We show that this Lagrangian mean curvature
flow can be continued for all time, through finite time singularities, and
converges to a chain of special Lagrangians, thus verifying various aspects of
Joyce's conjectures in this setting. We show that the singularities of the flow
are neck pinches in the sense conjectured by Joyce. We also give examples where
such finite time singularities are guaranteed to occur.
