  A scheme for rapidly and accurately computing solutions to boundary integral
equations (BIEs) on rotationally symmetric surfaces in R^3 is presented. The
scheme uses the Fourier transform to reduce the original BIE defined on a
surface to a sequence of BIEs defined on a generating curve for the surface. It
can handle loads that are not necessarily rotationally symmetric. Nystrom
discretization is used to discretize the BIEs on the generating curve. The
quadrature is a high-order Gaussian rule that is modified near the diagonal to
retain high-order accuracy for singular kernels. The reduction in
dimensionality, along with the use of high-order accurate quadratures, leads to
small linear systems that can be inverted directly via, e.g., Gaussian
elimination. This makes the scheme particularly fast in environments involving
multiple right hand sides. It is demonstrated that for BIEs associated with the
Laplace and Helmholtz equations, the kernel in the reduced equations can be
evaluated very rapidly by exploiting recursion relations for Legendre
functions. Numerical examples illustrate the performance of the scheme; in
particular, it is demonstrated that for a BIE associated with Laplace's
equation on a surface discretized using 320,800 points, the set-up phase of the
algorithm takes 1 minute on a standard laptop, and then solves can be executed
in 0.5 seconds.
