Hi Maple Primes

I have this parabola -

y2:=6z^2 +z+244;

x2:=9z^2+3z+367;

I use the eliminate command to write without z;

c:=eliminate([y=y2,x=x2],z);

Then I have the expression

e2 = 4x^2-12xy+9y^2-7x+9y+369.

My questions is, what integer values for x and y are on the curve e2?

I think the answer may be exactly when z is an integer.

How could I determine this given only e2?

Regards,
Matt