Boundary Element Method (BEM) integral solutions were developed for axisymmetric nonturbulent flows. These included two solutions to Poisson's vector potential equation that describe the kinematic aspects of Navier-Stokes flows. The Navier-Stokes solution was completed by solving the vorticity transport equation with planar finite difference representations modified to axisymmetric form. Three simple, laminar, axisymmetric flow problems were solved to verify the form of the BEM integrals and the accuracy of the numerical integration algorithms. A direct BEM integral solution to the axisymmetric potential equation was also derived but not developed or tested.