The kinematics of a potential vortex offers an interesting flow history for a rheologically complex material, and earlier work on that subject led us to consider the behavior of a Newtonian drop in three related time dependent flow fields [K. Sarkar, W.R. Schowalter, Deformation of a two-dimensional drop at non-zero Reynolds number in time-periodic extensional flows: numerical simulation, J. Fluid Mech., 2000, submitted for publication; K. Sarkar, W.R. Schowalter, Deformation of a two-dimensional viscous drop in time-periodic extensional flows: analytical treatment, J. Fluid Mech., 2000, submitted for publication]. In the work reported here the drop, characterized by an upper-convected Maxwell model (UCM), is suspended in an incompressible Newtonian fluid. Again, three related flows are considered. The first is that of a potential vortex, modeled by an extensional flow field near the drop with rotating axes of stretching. The second is a generalization of the first and is called rotating extensional (RE) flow, in which the frequency of revolution of the flow is varied independently of the shear rate. Finally, we consider oscillating extensional (OE) flow. Calculations were performed at small but non-zero Reynolds numbers using an ADI front-tracking/finite difference method. We have developed an analytic elastic-viscous stress splitting scheme obtained by an integration by parts of the constitutive equation. The scheme explicitly separates the diffusive part of the momentum equation for a wide range of differential constitutive relations. An ADI implementation is executed for the diffusive part. We investigate the effects of periodicity, Reynolds number and relaxation time on the drop dynamics. For a vortex and an RE flow, the long-time deformation reaches a steady value, and the drop attains a revolving, steady elliptic shape. The long-time values of deformation show complex non-monotonic behavior with variation in Weissenberg number, an effect of the decreased damping and increased elasticity, as well as the presence of a shear wave triggered by the UCM constitutive relation. The first two effects are modeled successfully by a simple ODE presented in Appendix A. The wave effects are briefly discussed in Appendix B.