The main result of the paper is the positive solution of the restricted Burnside problem for the groups with odd exponent, i.e. the author proves that the order of finite groups of odd exponent \(n\) with \(m\) generators is bounded by a function depending on \(m \)and \(n\) only. As in the well-known paper of A. I. Kostrikin on the restricted Burnside problem for groups of prime exponent the main tool is the study of associated Lie rings satisfying certain polynomial identities. So the main technical result of the paper (of great importance for the theory of Lie rings) says that a finitely generated Lie algebra over \(\mathbb{Z}_ p\) satisfying a linearized Engel identity and such that for some natural \(s\) and any commutator \(\rho\) in the generators one has \((ad\,\rho)^ s=0\) is nilpotent. There is a corollary: An Engel Lie ring is locally nilpotent.