We consider several second order fully nonlinear PDE in two variables and construct their exact solutions having the form of travelling waves. The corresponding ODE are solvable in elementary, hyperbolic, special functions or in quadratures. Therefore, the solutions can be globally defined, can blow up at the finite end point of a ray or at the end points of some finite interval. At some cases the separation of variables enables us to find explicit solutions. As the machinery of the ODE is applied the solutions are usually local. Then we find the radial solutions of the Liouville equation to the Cauchy, respectively Dirichlet problem in circular domains in the plane (circle, compliment to the circle, annulus). The maximum principle gives unique solution. Otherwise 0, 1, 2 solutions exist. The solvability of the constant data Dirichlet problem in the annulus is reduced to the solvability of two point boundary value problem for second order fully nonlinear PDE with exponential nonlinearity. Boundary value problems for Liouville type ODE possess applications in geometry, heat processes, chemistry, astronomy and in other sciences. We mention only that the Cauchy problem in the complement of a circle either has a global solution with logarithmic asymptote at infinity or blows up for some finite \( r_{0} \) .