This paper proposes an anti-windup design method for systems which contain two sources of nonlinearities, namely actuator saturation and a quadratic nonlinearity. Two classes of systems are considered: (1) the saturation nonlinearity is additive to the quadratic nonlinearity; (2) the saturation nonlinearity is multiplicative of a state-dependent matrix. A nonlinear dynamic inversion controller is augmented with an anti-windup compensator. The design of the anti-windup compensator is carried out in order to enlarge the region of attraction in the subspace of the plant variables. From theoretical conditions some convex optimization problems are proposed to compute the gains of the anti-windup compensator. Numerical examples illustrate the effectiveness of the proposed technique.