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.