Let \(k\in\mathbb Z\) and consider the system of equations \[ xyz = 1,\quad x + y + z = k. \] The solutions \(x,y,z\) of this system of equations were first considered by Cassels over \(\mathbb Q\). Several authors considered generalizations of this problem, extending the type of the equation or considering the solutions in a ring of algebraic integers instead of \(\mathbb Z\), cf. e.g. \textit{A. Bremner} [Acta Arith. 57, No. 4, 375--385 (1991; Zbl 0686.10012)]. This paper is the continuation of the authors' work [Acta Arith. 162, No. 4, 381--392 (2014; Zbl 1358.11049)] where the correspondng curve \[ {\mathcal E}_k : Y^2 = 1 - 2kX + k^2X^2 - 4X^3 \] was defined and the authors determined all solutions of the system of equations in algebraic integers \(x,y,z\) in a field of degree at most 4, with \(k\in\mathbb Z\) such that \(|{\mathcal E}_k(\mathbb Q)|=3\). In this work, the authors extend the results to include \(k =-1\) and \(k = 5\), and prove that this, then, solves the problem for all \(k\) with \({\mathcal E}_k(\mathbb Q)\) finite. In Section 2, they prove that for \(k\in\mathbb Z\), if \({\mathcal E}_k(\mathbb Q)\) is finite, but not of order 3, then \(k\in \{1,-5\}\). In Section 3, all solutions to the system of equations are found with \((x,y,z)\in {\mathcal O}_F^3\) where \([F : \mathbb Q] \leq 3\) and \(k\in \{1,-5\}\). Finally, in Section 4, we they solve the case \([F : \mathbb Q] = 4\).