Patterned thin gold lines when undergoing solid-state dewetting break up into linear grain sections interspersed with occasionally larger ”abacus” particles. We model this system using coupled geometric flows, where the exterior surfaces of the grains evolve via Surface Diffusion, V = −∆Γ(t)H, and the internal grain boundaries evolve via Mean Curvature Flow, V = AH. In this talk we first motivate the problem based on these experimental observations. Next, we describe the steady states for this mixed-order system. By assuming axi-symmetry, we can systematically construct composite equilibria by piecing together Delaunay surfaces—specifically, combining unduloidal, cylindrical and spherical exterior surfaces which are partitioned by planar or catenoidal internal grain boundaries. We then undertake a stability analysis by first linearising the coupled (nonlinear) problem about these composite steady states, deriving a Jacobi operator whose strict positivity implies asymptotic stability of the linearised problem. Using this approach and building upon classical Rayleigh stability criteria, we obtain stability/instability predictions for steady-state configurations consisting of near-cylindrical and near-spherical grains which join together along a planar grain boundary.