Recall:
(1) Given $A,B \in \mathbb{C}[x,y]$ we have: $\{A,B\}$ are algebraically independent over $\mathbb{C}$ if and only if $\operatorname{Jac}(A,B) \neq 0$.
(2) By definition, a Jacobian pair is a pair $C,D \in \mathbb{C}[x,y]$ satisfying $\operatorname{Jac}(C,D) \in \mathbb{C}-\{0\}$.
(3) Keller's theorem (Theorem 2.1(b)) says the following: Let $C,D \in \mathbb{C}[x,y]$ be a jacobian pair. If $\mathbb{C}(C,D)=\mathbb{C}(x,y)$, then $\mathbb{C}[C,D]=\mathbb{C}[x,y]$.
Let $u,v \in \mathbb{C}[x,y]$ be two algebraically independent elements over $\mathbb{C}$ satisfying $\operatorname{Jac}(u,v) \in \mathbb{C}[x,y]-\mathbb{C}$, so $\{u,v\}$ is not a Jacobian pair. Assume that $\mathbb{C}(u,v)=\mathbb{C}(x,y)$. Is it true that necessarily $u=x,v=xy$ or $u=y,v=xy$ (or vice versa)?
I have not succeeded to find other examples, but perhaps I am missing something.
In view of Mohan's comment, I wish to change my question to:
What can be said about such $u,v$? (some kind of a general form?).
Please also see this question and this question.
Any hints and comments are welcome!