  Characterizing existence or not of periodic orbit is a classical problem and
it has both theoretical importance and many real applications. Here, several
new criterions on nonexistence of periodic orbits of the planar dynamical
system $\dot x=y,~\dot y=-g(x)-f(x,y)y$ are obtained in this paper, and by
examples showing that these criterions are applicable, but the known ones are
invalid to them. Based on these criterions, we further characterize the local
topological structures of its equilibrium, which also show that one of the
classical results by A.F. Andreev [Amer. Math. Soc. Transl. 8 (1958), 183--207]
on local topological classification of the degenerate equilibrium is
incomplete. Finally, as another application of these results, we classify the
global phase portraits of a planar differential system, which comes from the
third question in the list of the 33 questions posed by A. Gasull and also from
a mechanical oscillator under suitable restriction to its parameters.
