In general this is a really hard problem (Diophantine equation)... I'm sure that if this is a real question from somewhere, there must be some trick to solving it, but I'm not sure what it is off the top of my head.
https://en.wikipedia.org/wiki/Diophantine_equation
I would start just by plotting the equation (iterate over x = -50 to 50, y = - 50 to 50 or something like that) and see if there's any sort of pattern. Of course, just graphical inspection doesn't prove that there isn't a really far-out number that also solves the equation.
It seems pretty chaotic to me, I don't see a clear pattern.
https://www.wolframalpha.com/input/?i=plot++round%28x%5E3+%2B+y%5E3+-+x%5E2+-+y%5E2+-+42+x+y%29+%3D+0+from+x+%3D+-40+to+40%2C+y+%3D+-40+to+40
EDIT: I didn't realize you said natural numbers. I was thinking all integers!
The equation looks clearly bounded by graphical inspection on the +x, +y quadrant.
https://www.wolframalpha.com/input/?i=plot++round%28x%5E3+%2B+y%5E3+-+x%5E2+-+y%5E2+-+42+x+y%29+%3D+0+from+x+%3D+0+to+50%2C+y+%3D+0+to+50
What I said still applies -- just graphical inspection doesn't prove it, but it looks pretty solid.
So all you need to do is iterate over a bounded range of positive {x, y} pairs, see which ones solve the equation.
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Since we're just working with positives, it becomes clear after I stare at the equation for a bit that the x^3 + y^3 term will eventually beat the x^2 + y^2 + 42xy term, but initially the RHS is greater than the LHS due to the large constant factor (42). Still that's no where near a proof.