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 $\op{max}\{a, b\}\leq 50000$.