We present Free-HeadGAN, a person-generic neural talking head synthesis system. We show that modeling faces with sparse 3D facial landmarks are sufficient for achieving state-of-the-art generative performance, without relying on strong statistical priors of the face, such as 3D Morphable Models. Apart from 3D pose and facial expressions, our method is capable of fully transferring the eye gaze, from a driving actor to a source identity. Our complete pipeline consists of three components: a canonical 3D key-point estimator that regresses 3D pose and expression-related deformations, a gaze estimation network and a generator that is built upon the architecture of HeadGAN. We further experiment with an extension of our generator to accommodate few-shot learning using an attention mechanism, in case more than one source images are available. Compared to the latest models for reenactment and motion transfer, our system achieves higher photo-realism combined with superior identity preservation, while offering explicit gaze control.