Let $(K,v)$ be a henselian valued field, with valuation ring $\mathcal{O}$ and residue field $Kv$.
Then given a polynomial $f \in \mathcal{O}[x]$, henselianity tells that given some suitable non-degeneracy condition, you can lift a zero of $\overline{f}$ in $Kv$ to a zero of $f$ in $\mathcal{O}$. Here $\overline{f}$ denotes the reduction of $f$ modulo the maximal ideal.
I know there are generalizations of this which look at $n$ functions in $n$ variables, and the non-degeneracy condition comes from the Jacobian determinant. I'm interested in a slightly different extension:
$\bf{Question}$: Suppose $f(x,y)$ is a polynomial in two (or more) variables over $\mathcal{O}$. Can one specify some non-degeneracy condition which guarantees that solutions from the residue field lift?