  This paper is a continuation of the study of spectral flow inside essential
spectrum initiated in \cite{AzSFIES}. Given a point $\lambda$ outside the
essential spectrum of a self-adjoint operator $H_0,$ the resonance set,
$\mathcal R(\lambda),$ is an analytic variety which consists of self-adjoint
relatively compact perturbations $H_0+V$ of $H_0,$ for which $\lambda$ is an
eigenvalue. One may ask for criteria for the vector $V$ to be tangent to the
resonance set. Such criteria were given in \cite{AzSFnRI}.
  In this paper we study similar criteria for the case of $\lambda$ inside the
essential spectrum of $H_0.$ For the case $\lambda \in \sigma_{ess}(H_0)$ the
resonance set is defined in terms of the well-known limiting absorption
principle. Among the results of this paper is that the resonance set contains
plenty of straight lines, moreover, given any regular relatively compact
perturbation $V$ there exists a finite rank self-adjoint operator, $\tilde V,$
such that the straight line $H_0 + \mathbb R(V-\tilde V)$ belongs to the
resonance set.
  Another result of this paper is that inside the essential spectrum there
exist plenty of transversal to the resonance set perturbations $V$ which have
order $\geq 2,$ in contrast to what happens outside the essential spectrum,
\cite{AzSFnRI}.
