5.56 (1976)

b) Yes, there does. Every non-soluble variety $\mathfrak{V}$ contains a non-trivial group coinciding with the derived subgroup, the direct limit of the spectrum $F \xrightarrow{\varphi} F \xrightarrow{\varphi} \dots$, where $F$ is the free group in $\mathfrak{V}$ on the free generators $x_i, a_2, \dots$ and $\varphi$ is the homomorphism given by $x_i \to [x_{2i-1}, x_{2i}]$. As shown in (Yu. P. Razmyslov, Algebra and Logic, 10 (1971), 21–29) the Kostrikin variety of locally nilpotent groups of prime exponent $p \geqslant 5$ is unsoluble. (E. I. Khukhro, I. V. L’vov, Letter of June, 19, 1976.) The same was also proved in (Yu. A. Kolmakov, Math. Notes, 35, no. 5-6 (1984), 389–391). An example answering the question was also produced in (M. R. Vaughan-Lee, J. Wiegold, Bull. London Math. Soc., 13, no. 1 (1981), 45–46).