  A Li\'enard type nonlinear oscillator of the form
$\ddot{x}+kx\dot{x}+\frac{k^2}{9}x^3+\lambda_1 x=0$, which may also be
considered as a generalized Emden type equation, is shown to possess unusual
nonlinear dynamical properties. It is shown to admit explicit nonisolated
periodic orbits of conservative Hamiltonian type for $\lambda_1>0$. These
periodic orbits exhibit the unexpected property that the frequency of
oscillations is completely independent of amplitude and continues to remain as
that of the linear harmonic oscillator. This is completely contrary to the
standard characteristic property of nonlinear oscillators. Interestingly, the
system though appears deceptively a dissipative type for $\lambda_1\leq0$ does
admit a conserved Hamiltonian description, where the characteristic decay time
is also independent of the amplitude. The results also show that the criterion
for conservative Hamiltonian system in terms of divergence of flow function
needs to be generalized.
