(all joint work with Jason Lotay) A standing conjecture of Richard Thomas, motivated by mirror symmetry, gives a stability condition supposed to control the existence of a special Lagrangian submanifold in a given Hamiltonian isotopy class of Lagrangians. Later, Thomas and Yau conjectured a similar stability condition controls the long-time existence of the Lagrangian mean-curvature flow. In this talk I will explain how Jason Lotay and myself have recently proved versions of these conjectures on all circle symmetric hyperKahler 4-manifolds.