How many N = 4 strings exist?

Possible ways of constructing new extended fermionic strings with non-linear N=4 world-sheet supersymmetry are considered. N=4 string theory constraints are required to form a quasi-superconformal algebra, and have conformal dimensions . The most general non-linear N=4 quasi-superconformal algebra is , whose linearization is the so-called 'large' N=4 superconformal algebra. The algebra has affine Lie component, and . We construct a quantum BRST charge for this algebra, and show that it is nilpotent only if . This restricts the previously proposed continuous -family of critical N=4 strings to only one theory based on the algebra which is isomorphic to the SO(4)-based Bershadsky - Knizhnik quasi-superconformal algebra. We propose the (non-covariant) Hamiltonian action for the new N=4 string theory. Our results imply the existence of two different critical N=4 fermionic string theories: the 'old' one based on the 'small' linear N=4 superconformal algebra and having the total ghost central charge , and the new one with the non-linearly realized N=4 supersymmetry, based on the SO(4) quasi-superconformal algebra and having . Both critical N=4 string theories have negative 'critical dimensions' and do not admit unitary matter representations.