Construction of initial vortex-surface fields and Clebsch potentials for flows with high-symmetry using first integrals

We develop a systematic methodology to construct the explicit, general form of vortex-surface fields (Yang and Pullin, J. Fluid Mech., 661, 2010) and Clebsch potentials based on first integrals of the characteristic equation of a given three-dimensional velocity-vorticity field. This methodology is successfully applied to the initial fields with the zero helicity density and high symmetry, e.g., initial fields with the Taylor-Green and the Kida-Pelz symmetries.