A warm-up:

$$ \left(\frac{s^3-t^3}2\right)^2 +\ (s\cdot t)^3\ =\ 
        \left(\frac{s^3+t^3}2\right)^2 $$

provides integer solutions for $\,\ s\equiv t \mod 2 $.

*(Sorry, I couldn't help it).*