Let $V$ be a vector subspace of the associative algebra $A$ over a field $F$ which generates the algebra $A$. In the paper under review the author studies the identities of the pair $(A,V)$, i.e., the polynomials $f(x\sb{1},\ldots,x\sb{n})$ in the free associative algebra $F\langle X\rangle$ such that $f(v\sb{1},\ldots,v\sb{n})=0$ in $A$ for all $v\sb{1},\ldots,v\sb{n}\in V$. By analogy with the theory of varieties of algebras with polynomial identities, one considers the finite basis problem (or the Specht problem) for varieties of pairs. The first result of the paper states that over a field of characteristic 0 the identities of any pair satisfying the identity $x[y,z]=0$ follow from a finite number of identities. A similar result holds for the identity $[x,y]z=0$. When the base field $F$ is infinite and of arbitrary characteristic the finite basis property for these two identities is established for identities in a bounded number of variables. The second result of the paper states that over an infinite field the union of the varieties of pairs defined by the identities $x[y,z]=0$ and $[x,y]z=0$, respectively, is not finitely based.