  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 Schr\"odinger 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
(ANNNS) model at its Lifshitz points of first and second order.
