  We propose a method for tracing implicit real algebraic curves defined by
polynomials with rank-deficient Jacobians.
  For a given curve $f^{-1}(0)$, it first utilizes a regularization technique
to compute at least one witness point per connected component of the curve.
  We improve this step by establishing a sufficient condition for testing the
emptiness of $f^{-1}(0)$.
  We also analyze the convergence rate and carry out an error analysis for
refining the witness points.
  The witness points are obtained by computing the minimum distance of a random
point to a smooth manifold embedding the curve while at the same time
penalizing the residual of $f$ at the local minima.
  To trace the curve starting from these witness points, we prove that if one
drags the random point along a trajectory inside a tubular neighborhood of the
embedded manifold of the curve, the projection of the trajectory on the
manifold is unique and can be computed by numerical continuation.
  We then show how to choose such a trajectory to approximate the curve by
computing eigenvectors of certain matrices.
  Effectiveness of the method is illustrated by examples.
