A sextic diophantine chain and a related Mordell curve

In this paper we obtain parametric as well as numerical solutions of the sextic diophantine chain $\varphi (x_1,y_1,z_1)=\varphi (x_2,y_2,z_2)=\varphi (x_3,y_3,z_3)=k$ where $\varphi (x,y,z)$ is a sextic form defined by $\varphi (x,y,z)$ $=x^6+y^6+z^6-2x^3{y}^3-2x^3{z}^3-2y^3{z}^3$ and $k$ is an integer. Each numerical solution of such a sextic chain yields, in general, nine rational points on the Mordell curve $y^2=x^3+k\mathrm{/}4$. While all of these nine points are not independent in the group of rational points of the Mordell curve, we have constructed a parameterized family of Mordell curves of generic rank $\ge 6$ using the aforementioned parametric solution of the sextic diophantine chain. Similarly, the numerical solutions of the sextic chain yield additional examples of Mordell curves whose rank is $\ge 6$.