Diophantine equations play a central role in number theory, focusing on integer solutions of polynomial equations. In particular, homogeneous cubic Diophantine equations in four variables have attracted considerable attention due to their rich algebraic structure and diverse solution behavior. This paper investigates the homogeneous cubic equationx3+y3=42zw2x³ + y³ = 42zw²x3+y3=42zw2. Motivated by recent work in which only a few sets of integer solutions were obtained, we construct new families of integer solutions exhibiting structured and parametric patterns. Using suitable algebraic manipulations and parameterization techniques, we derive infinitely many non-trivial integer solutions of the given equation. The obtained results extend previously known solutions and reveal additional symmetric relationships among the variables. Consequently, this study contributes further insight into the distribution of integer solutions for higher-degree Diophantine equations in multiple variables
Alexander Müller (Mon,) studied this question.