There is considered the differential equation
with , an integer, is a higher order term and a small perturbation term. Let , where and is the strip domain defined by . Let and be real analytic on and analytic quasi-periodic in with vector frequency . If , the equation has 0 as degenerate equilibrium. It is proved that, when is sufficiently small, the differential equation can be reduced via an affine quasi-periodic transformation to a suitable normal form with 0 as equilibrium so it has a quasi-periodic solution near 0.