Summary: Let \(A\) be a commutative domain of characteristic 0 which is finitely generated over \(\mathbb{Z}\) as a \(\mathbb{Z}\)-algebra. Denote by \(A^*\) the unit group of \(A\) and by \(\overline{K}\) the algebraic closure of the quotient field \(K\) of \(A\). We shall prove effective finiteness results for the elements of the set \[ \mathcal{C}:=\{ (x,y)\in (A^*)^2 \mid F(x,y)=0 \} \] where \(F(X,Y)\) is a non-constant polynomial with coefficients in \(A\) which is not divisible over \(\overline{K}\) by any polynomial of the form \(X^{m}Y^{n}-\alpha\) or \(X^{m}-\alpha Y^{n}\), with \(m, n\in\mathbb{Z}_{\geq0}\), \(\max(m,n)>0\), \(\alpha\in\overline{K}^*\). This result is a common generalisation of effective results of J.-H. Evertse and K. Győry [Math. Proc. Camb. Philos. Soc. 154, No. 2, 351–380 (2013; Zbl 1332.11040)] on \(S\)-unit equations over finitely generated domains, of E. Bombieri and W. Gubler [Heights in Diophantine geometry. Cambridge: Cambridge University Press (2006; Zbl 1115.11034)] on the equation \(F(x, y) = 0\) over \(S\)-units of number fields, and it is an effective version of S. Lang’s general but ineffective theorem [Publ. Math., Inst. Hautes Étud. Sci. 6, 319–335 (1960; Zbl 0112.13402)] on this equation over finitely generated domains. The conditions that \(A\) is finitely generated and \(F\) is not divisible by any polynomial of the above type are essentially necessary.