Asymptotically shearfree congruences in (2,2) spacetimes and Burgers' equation

The paper proves that any asymptotically shearfree congruence at the conformal infinity (scri) in a (2,2)-signature spacetime is determined locally by a solution to the pair of forced inviscid Burgers’ equations L_u+LL_x={\sigma}(u,x,y,L) and M_u+MM_y={\sigma}‘(u,x,y,M) where u,x,y are Bondi coordinates of scri. The functions {\sigma} and {\sigma}’ are determined naturally by the projective structure on the {\alpha} and {\beta} surfaces that foliate scri.