Beginning with the work of Dean in 1927, regular perturbation methods have been used to study flows of incompressible Newtonian, generalized Newtonian and viscoelastic fluids in curved pipes of circular cross section. In these studies, the perturbation parameter is the curvature ratio: the cross sectional radius of the pipe divided by the radius of curvature of the pipe centerline. Closed form perturbation solutions for the equations of motion for second order fluids have previously been obtained by other authors for the special case when the second normal stress coefficient is zero, H.G. Sharma, A. Prakash, Ind. J. Pure Appl. Math 8 (1977) 546-557; P.J. Bowen, Ph.D. Thesis, University of Wales, (1990); P.J. Bowen, A.R. Davies, K. Waiters, J. Non-Newtonian Fluid Mech. 38 (1991) 113-126. Here, we obtain closed form solutions for the perturbation equations even when the second normal stress difference is non-zero. We show that for a countable number of combinations of non-dimensional parameters a perturbation solution exists but is not unique. For other combinations of parameters a perturbation solution does not even exist. This latter result implies that for these parameter values there does not exist a steady, fully developed solution for flow in curved pipes which is a perturbation of the straight pipe solution, regardless of the magnitude of the curvature ratio. We emphasize that these singular points do not arise when the second normal stress coefficient is zero. A solution to the perturbation equations exists and is unique for values of the material constants which correspond to real polymeric fluids. For these values of the material constants, the secondary motion at zero Reynolds number is qualitatively similar to that arising in Newtonian fluids due to inertial effects. (C) 2000 Elsevier Science B.V. All rights reserved.