  It is demonstrated that the known for a long time transition between the gap
and the gapless states in the Abrikosov-Gor'kov theory of a superconductor with
paramagnetic impurities is of the Lifshitz type, i.e. of the $2\frac12$ order
phase transition. We reveal the emergence of a cuspidal edge at the density of
states surface $N(\omega,\Delta_0)$ ($\Delta_0$ is the value of the
superconducting order parameter in the absence of magnetic impurities) and the
occurrence of the catastrophe phenomenon at the transition point. We study the
stability of such a transition with respect to the spatial fluctuations of the
magnetic impurities critical concentration $n_s$ and show that the requirement
for validity of its mean field description is unobtrusive: $\nabla \left( {\ln
{n_s}} \right) \ll \xi^{-1} $ (here $\xi$ is the superconducting coherence
length). Finally, we show that, similarly to the Lifshitz point, the $2\frac12$
order phase transition should be accompanied by the corresponding
singularities. For instance, the superconducting thermoelectric effect has a
giant peak exceeding the normal value of the Seebeck coefficient by the ratio
of the Fermi energy and the superconducting gap. The concept of the experiment
for the confirmation of $2\frac12$ order transition nature is proposed. The
obtained theoretical results can be applied for the explanation of recent
experiments with lightwave-driven gapless superconductivity, for the new
interpretation of the disorder induced transition $s_{\pm}$-$s_{++}$ states via
gapless state in multi-band superconductors, for better understanding of the
gapless color superconductivity in quantum chromodynamics, the string theory.
