The set of solutions of quadratic equation $a^2+b^2=c^2$ on $\mathbb{Z}$ can be described by Pythagorean triples up to multiplication. Can I use similar results on the ring of integer coefficient polynomials $\mathbb{Z}[x]$? More concretely,
(1) Is there a complete description of the set of solutions of $a^2+b^2=c^2$, $a,b,c\in\mathbb{Z}[x]$?
(2) In general, is there a theory on the class of equations $a^2+f(x)b^2=c^2$, where $f(x)\in\mathbb{Z}[x]$ is a given polynomial?
Pythagorean triples in a polynomial ring $R[x]$ can be classified as follows (see Theorem $5.1$ in K. Conrad's notes here):
Theorem $5.1$: All primitve Pythagorean triples $(f(t),g(t),h(t))$ with $f^2+g^2=h^2$ are given by the formulas \begin{align*} f(t) & = c(k(t)^2-\ell(t)^2),\\ g(t) & = \pm 2c\cdot k(t)\ell(t),\\ h(t) & = \pm c(k(t)^2+\ell(t)^2), \end{align*} where $c\in R^{\times}$ and $k(t),\ell(t)$ are two relative prime polynomials in $R[t]$.