Unique jet determination and extension of germs of CR maps into spheres

We provide a new way of simultaneously parametrizing arbitrary local CR maps from real-analytic generic manifolds M\subset \mathbb{C}^N into spheres \mathbb{S}^{2N'-1}\subset \mathbb{C}^{N'} of any dimension. The parametrization is obtained as a composition of universal rational maps with a holomorphic map depending only on M. As applications, we obtain rigidity results of different flavours such as unique jet determination and global extension of local CR maps.