On primitive integer solutions of the Diophantine equation $x^3\pm y^3=a^k\pm b^k$

In this note we consider the title Diophantine equation from both theoretical as well as experimental point of view. In particular, we prove that for $k=4,6$ and each choice of the signs our equation has infinitely many co-prime positive integer solutions. For $k=5,7$ and all choices of the signs we computed all co-prime positive integer solutions $(x,y,a,b)$ satisfying the condition $\text{op}max\{a,b\}\le 50000$.