  In this paper we study the linear Weingarten equation defined by the fully
non-linear PDE
  $$a\, \mbox{div}\frac{Du}{\sqrt{1+|Du|^2}}+b\,
\frac{\mbox{det}D^2u}{(1+|Du|^2)^2}=\phi\left(\frac{1}{\sqrt{1+|Du|^2}}\right)$$
in a domain $\Omega\subset\mathbb{R}^2$, where $\phi\in C^1([-1,1])$ and
$a,b\in\mathbb{R}$. We approach the existence of radial solutions when $\Omega$
is a disk of small radius, giving an affirmative answer when the PDE is of
elliptic type. In the hyperbolic case we show that no radial solution exists,
while in the parabolic case we find explicitly all the solutions. Finally, in
the elliptic case we prove uniqueness and symmetry results concerning the
Dirichlet problem of such equation.
