I know one form of them as dQ/dt +dE/dx +dF/dy=0 where Q,E,F are 4X1 matrices.
Look up Burnett in Chapter 15, Chapman & Cowling, "The mathematical theory of non-uniform gases", 3rd edition, CUP [1970].
For the different forms of Burnett equations I would recommend the following paper:
Agarwal et al
Beyond Navier-Stokes: Burnett equations for flows in the continuum-transition regime
Physics of Fluids, Vol 13(10), pp. 3061-3085, Oct. 2001
Burnett equations are obtained using a Chapman-Enskog expansion of Boltzmann equations. Then you need a closure to solve the Burnett equations. For Navier-Stokes the closure is the Stokes relation, for Burnett you can not use it. As a consequence there are as many sets of Burnett equations than ways to close them.
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