  For a nondegenerate analytic system with a conserved quantity, a classic
result by Lyapunov guarantees the existence of an analytic manifold of periodic
orbits tangent to any two-dimensional, elliptic eigenspace of a fixed point
satisfying nonresonance conditions. These two dimensional manifolds are
referred as Lyapunov Subcenter Manifolds (LSM).
  Numerical and experimental observations in the nonlinear vibrations
literature suggest that LSM's often persist under autonomous, dissipative
perturbations. These perturbed manifolds are useful since they provide
information of the asymptotics of the convergence to equilibrium.
  In this paper, we formulate and prove precise mathematical results on the
persistence of LSM under dissipation.
  We show that, under mild non-degeneracy conditions on the perturbation, for
small enough dissipation, there are analytic invariant manifolds of the
perturbed system that approximate (in the analytic sense) the LSM in a fixed
neighborhood. We also provide examples that show that some non-degeneracy
conditions on the perturbations are needed for the results to hold true.
  We also study the dependence of the manifolds on the dissipation parameter.
If $\varepsilon$ is the dissipation parameter, we show that the manifolds are
real analytic in $(-\varepsilon_0, \varepsilon_0) \setminus \{0\} $ and
$C^\infty $ in $(-\varepsilon_0, \varepsilon_0)$. We construct explicit
asymptotic expansions in powers of $\varepsilon$ (which presumably do not
converge).
  Finally, we present applications of our results to several mechanical
systems.
