We have

${\mathrm{x}}^2{\mathrm{y}}^2-9{\mathrm{y}}^2-6{\mathrm{x}}^2\mathrm{y}+54\mathrm{y}=0$

or ${\mathrm{y}}^2({\mathrm{x}}^2-9)-6\mathrm{y}({\mathrm{x}}^2-9)=0$

or $\mathrm{y}(\mathrm{y}-6)(\mathrm{x}-3)(\mathrm{x}+3)=0$

or $\mathrm{y}=0,\mathrm{y}=6,\mathrm{x}=3,\mathrm{x}=-3$

So, the given equation represents four straight lines which form a square