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Let k ∈ ℤ be such that ${|}_k\left(\mathbb{Q}\right)|=3$, where ${}_k:y²=1-2kx+k²x²-4x³$. We determine all solutions to xyz = 1 and x + y + z = k in integers of number fields of degree at most four over ℚ.

Let k ∈ ℤ be such that ${|}_k\left(\mathbb{Q}\right)|$ is finite, where ${}_k:y²=1-2kx+k²x²-4x³$. We complete the determination of all solutions to xyz = 1 and x + y + z = k in integers of number fields of degree at most four over ℚ.