[For part I see ibid. 5, 123-131 (1985; Zbl 0606.16023).] The author proves commutativity of an associative ring \(R\) satisfying one of the following conditions: (1) for each \(x,y\in R\) there exists a co-monic polynomial \(p(t)\in tZ[t]\), such that \([x,y]=[x,y](p(xy)-p(yx))\); (2) for each \(x,y\in R\) there exist \(p(t),q(t)\in tZ[t]\) with \(q(t)\) of the form \((t-t^2f(t))\), such that \([x,y]=[x,y]p(x)q(y)\); (3) \((R,+)\) is a torsion group and for each \(x,y\in R\) there exist \(p(t),q(t)\in tZ[t]\) such that \([x,y]=[x,y]p(x)q(y)\) and the coefficients of \(q'(t)\) have highest common factor 1; (4) for each \(x,y\in R\) there exists \(n=n(x,y)\geq 1\) for which \([y,x]=[y^mx^n,x]\), where \(m\) is a fixed positive integer; (5) for each \(x,y\in R\) either \([x,y]=0\) or there exists an integer \(n=n(x,y)>1\) for which \(xy=x^ny^n\) and \(yx=y^nx^n\).