No. of integer solutions to a quadratic eq with three variab

Question

Here is the equation: $$z=18*18*18x^2 y^2+18*14*18x^2 y+ 18*49x^2+ 18*25y^2+ 18*10*18*18xy^2+ 36*71xy+37*7x+ 360y+71$$ I know it has obvious solution like $(0,0,71)$ for $(x,y,z)$. Have also found other solutions, like $(0,1,881)$ for $(x,y,z)$. I wonder if there is a general approach to show that there are many solutions of the form ???