In this paper some infinitely based varieties of groups are constructed and these results are transferred to the associative algebras (or Lie algebras) over an infinite field of an arbitrary positive characteristic.

In this paper we deal with the class ${\mathcal{E}}_k^{*}$ of groups $G$ for which whenever we choose two infinite subsets $X$, $Y$ there exist two elements $x\in X$, $y\in Y$ such that $\left[x,\underset{k}{\underbrace{y,\mathrm{\dots },y}}\right]=1$. We prove that an infinite finitely generated soluble group in the class ${\mathcal{E}}_k^{*}$ is in the class ${\mathcal{E}}_k$ of $k$-Engel groups. Furthermore, with $k=2$, we show that if $G\in {\mathcal{E}}_2^{*}$ is infinite locally soluble or hyperabelian group then $G\in {\mathcal{E}}_2$.