equation

It does not even have real solutions let alone integer ones. Are you question reads this way?

Let us compare the given eqn. with that of the eqn. of conics -- i.e. ,

with the eqn. ax² + 2hxy + by² + 2gx + 2fy + c = 0

We get a = 3 , h = -1 , b = 7 , g =0 , f= 0 , c= 0.

Here , abc + 2fgh -af² - bg² - ch² = 0

So this a degenrate conic .

Now , from the property of degenerate conics , if

ab - h² >0 ,

then the eqn. gives only a real point.

Here ab - h² = 7 x 3 - (-1)² > 0

Hence , the eqn. represents only a real point , or in other words , only a real solution.

To find that point , let us take the partial derivatives .

Partial derivative w.r.t x gives , 6x - 2y = 0 ----------- first equation

and partial derivative w.r.t y gives , 14y - 2x = 0 ------------------- second equation

So we get x = y / 3 from the first equation , and x = 7y from the second eqn .

these have one and only one unique solution ------ x = 0 , y = 0

Hence the given eqn. has only one unique solution -- x = 0 , y = 0 .