Summary: Let \(K\) be a number field, let \(S\) be a finite set of places of \(K\) containing the archimedean places and let \(\mu\), \(\alpha_1,\alpha_2,\alpha_3\) be non-zero elements in \(K\). Denote by \(\mathcal O_S\) the ring of \(S\)-integers in \(K\) and by \(\mathcal O_S^\times\) the group of \(S\)-units. Then the set of equivalence classes (namely, up to multiplication by \(S\)-units) of the solutions \(x,y,z,\varepsilon_1, \varepsilon_2,\varepsilon_3,\varepsilon\in\mathcal O_S^3\times(\mathcal O_S^\times)^4\) of the Diophantine equation
\[ (X-\alpha_1E_1Y)(-X\alpha_2E_2Y)(X-\alpha_3E_3Y)Z=\mu E, \]
satisfying \(\text{Card}\{\alpha_1\varepsilon_1,\alpha_2\varepsilon_2,\alpha_3\varepsilon_3\}= 3\), is finite. With the help of this last result, we exhibit new families of Thue-Mahler equations having only trivial solutions. Furthermore, we produce an effective upper bound for the number of these solutions. The proofs of this paper rest heavily on Schmidt’s subspace theorem.