Based on the fact that all three of your equations are equal to whole numbers, I would GUESS that all three numbers are whole numbers. If they aren't, I find it hard to believe that summing them, summing their squares, and summing their cubes all result in whole numbers.

Also, based on the fact that when you sum their cubes you get a smaller answer than when you sum their squares, one of them (at least) must be negative - the negative sign will go away when you have an even power (like 2) but will be present when you have an odd power (like 3), thus making the sum of odd powers less than the sum of even powers.

However, assuming that all that is correct, I tried all combinations of x, y, and z, with x ranging from -20 to 20 and y ranging from -20 to 20 (and z dependent on x and y, based on the first equation) and got no solutions.

Not sure if any of this is any help. Maybe it means they answers are outside of that range, which seems very doubtful (the lowest values I got for the sums of squares and cubes where when, obviously, x and y were close to 0). Maybe it means that the answers aren't whole numbers. Maybe it means there is no solution, which is kind of the direction I'm leaning, but I have no proof.