x²z + y²z + 4xy = 40,

x² + y² + xyz = 20

Hello!

The above posts were certainly enlightening.  I had convinced myself that I had found an infinite number of solutions to this problem, but only after looking at what others had posted did I realize my error.  So, almost all of my "solutions" had to be discarded, as they actually didn't satisfy the original equations.  However, I did end up with 4 possible solution triples.  Two of them have been mentioned previously, namely

(1,1,18) and (-1,-1,18).

There is another pair of solutions though!  If you are keen to try and find them on your own, then don't read any further, as I will give them away below.

The other pair is (2,2,3) and (-2,-2,3).  As far as I could tell in my calculations I exhausted all other possibilities.  So I believe that these are the only 4 answers.  It will certainly be interesting to see any more correspondence on this problem!  Thanks to Bractals for the great puzzle.

-John