In this paper we present results about ergodicity of dynamical systems on 2-adic spheres for 1-Lipschitz maps f : Z(2) -> Z(2) announced in [8], and extension of Theorem 3 from [8] for the case of spheres of radii greater than 1/8. We propose a new approach to study ergodic properties of 1-Lipschitz transformations of 2-adic spheres. We use a representation of continuous functions f via its van der Put series. This technique allows us to go beyond the classes of smooth 1-Lipschitz transformations which were studied earlier.