Infinite Number of Integers which are Sum of 3 Sixth Powers in 2 Ways

Theorem
There exist an infinite number of positive integers which can be expressed as the sum of $3$ sixth powers in $2$ different ways.

Proof
There are many parametric solutions to $x^6 + y^6 + z^6 = u^6 + v^6 + w^6$. One is given by:

This set of solutions also satisfy: