Extending one pair of solutions to other pairs of solutions in a polynomial function

I have:

$$f(x,y)=yx^4+a(y)x^3+b(y)x^2+c(y)x+d(y)$$ with:

$$a(y)=-(4y^2+6y+1)$$ $$b(y)=6y^3+18y^2+11y+6$$ $$c(y)=-(4y^4+18y^3+22y^2+6y+11)$$ $$d(y)=y^5+6y^4+11y^3+6y^2+6$$

I’m looking for $$x,y\in\Bbb Z^+ \text{ that solve } f(x,y)=0$$

From how I formulated this function, I know that $$f(11,6)=0$$, and I’m wondering if there are any extension methods that could be used to find other solutions if they exist.