Local scale invariance and strongly anisotropic equilibrium critical systems

A new set of infinitesimal transformations generalizing scale invariance for strongly anisotropic critical systems is considered. It is shown that such a generalization is possible if the anisotropy exponent theta=2/N, with N = 1,2,3.... Differential equations for the two-point function are derived and explicitly solved for all values of N. Known special cases are conformal invariance (N=2) and Schrodinger invariance (N=1). For N=4 and N=6, the results contain as special cases the exactly known scaling forms obtained for the spin-spin correlation function in the axial next-nearest-neighbor spherical model at its Lifshitz points of first and second order.