  We classify all rational maps $H \in K(x)^n$ for which ${\rm trdeg}_K
K(tH_1,tH_2,\ldots,tH_n) \le 2$, where $K$ is any field and $t$ is another
indeterminate.
  Furthermore, we classify all such maps for which additionally $JH \cdot H =
{\rm tr} JH \cdot H$ (where $JH$ is the Jacobian matrix of $H$), i.e. $$
\sum_{i=1}^n H_i \frac{\partial}{\partial x_i} H_k = \sum_{i=1}^n H_k
\frac{\partial}{\partial x_i} H_i $$ for all $k \le n$. This generalizes a
theorem of Paul Gordan and Max N\"other, in which both sides and the
characteristic of $K$ are assumed to be zero.
  Besides this, we use some of our tools to obtain several results about
$K$-subalgebras $R$ of $K(x)$ for which ${\rm trdeg}_K L = 1$, where $L$ is the
fraction field of $R$.
  We start with some observations about to what extent, L\"uroth's theorem can
be generalized.
