  The interplay of Anderson localization and electron-electron interactions is
known to lead to enhancement of superconductivity due to multifractality of
electron wave functions. We develop the theory of multifractally-enhanced
superconducting states in two-dimensional systems in the presence of spin-orbit
coupling. Using the Finkel'stein nonlinear sigma model, we derive the modified
Usadel and gap equations that take into account renormalizations caused by the
interplay of disorder and interactions. Multifractal correlations induce energy
dependence of the superconducting spectral gap. We determine the
superconducting transition temperature and the superconducting spectral gap in
the case of Ising and strong spin orbit couplings. In the latter case the
energy dependence of superconducting spectral gap is convex whereas in the
former case (as well as in the absence of spin-orbit coupling) it is concave.
Multifractality enhances not only the transition temperature but, in the same
way, the spectral gap at zero temperature. Also we study mesoscopic
fluctuations of the local density of states in the superconducting state.
Similarly to the case of normal metal, spin-orbit coupling reduce the amplitude
of fluctuations.
