Exact steady equilibrium solutions based on complex variable methods are presented describing the transport of both soluble and insoluble surfactant along an infinite interface between a constant pressure region and a deep viscous fluid, subjected to a straining flow in the far-field. The effect of surface diffusion on the formation of “stagnant caps” of surfactant is examined mathematically. In the insoluble case the underlying singularity structure of the explicit solution is analysed using exponential asymptotics, providing a classic illustration of the nonlinear Stokes phenomenon that has arisen from a physically relevant problem.