  We consider the lagrangian $L=F(R)$ in classical (=non-quantized)
two-dimensional fourth-order gravity and give new relations to Einstein's
theory with a non-minimally coupled scalar field. We distinguish between
scale-invariant lagrangians and scale-invariant field equations. $L$ is
scale-invariant for $F = c_1 R\sp {k+1}$ and a divergence for $F=c_2 R$. The
field equation is scale-invariant not only for the sum of them, but also for
$F=R\ln R$. We prove this to be the only exception and show in which sense it
is the limit of $\frac{1}{k} R\sp{k+1}$ as $k\to 0$. More generally: Let $H$ be
a divergence and $F$ a scale-invariant lagrangian, then $L= H\ln F $ has a
scale-invariant field equation. Further, we comment on the known generalized
Birkhoff theorem and exact solutions including black holes.
