The famous result of \textit{E. I. Zel'manov} [Izv. Akad. Nauk. SSSR, Ser. Mat. 54, No. 1, 42-59 (1990; Zbl 0704.20030)], solving the restricted Burnside problem, combined with a result of Kovács, given in [\textit{H. Neumann}, Varieties of groups, Springer-Verlag (1967; Zbl 0251.20001)], shows that the class, \({\mathcal R}_{p^\lambda}\), of locally finite groups satisfying the law \(X^{p^\lambda}=1\) is a variety. However, in general this variety is not nilpotent. In fact, for any subset \(W\) of \(F_\infty''\) (where \(F_\infty\) is the free group of countable rank) the variety \({\mathcal R}_{p^2}\cap{\mathcal V}(W)\) is not nilpotent. The main result of this paper is a converse of this: Theorem. Let \(W\) be a subset of \(F_\infty\). Then, if \(W\) is not contained in \(F_\infty''\), there exist integers \(\pi\), \(\mu\) such that, for all primes \(p>\pi\), each locally finite \(p\)-group in \({\mathcal V}(W)\) is nilpotent of class at most \(\mu\). -- A second theorem gives a more precise result.