Constants of motion in five component models of two dimensional turbulence

A system of equations derived by truncating a Fourier representation of the two-dimensional Euler equations using five frequencies is investigated. Five vectors are constructed which satisfy a pair of triad interactions. The general form of constants of motion in terms of the five frequencies is presented. The energy and enstrophy have been expressed as linear combinations of quadratic integrals and also have been expressed as functions of vorticity.