  We derive $C^{1,\sigma}$-estimate for the solutions of a class of non-local
elliptic Bellman-Isaacs equations. These equations are fully nonlinear and are
associated with infinite horizon stochastic differential game problems
involving jump-diffusions. The non-locality is represented by the presence of
fractional order diffusion term and we deal with the particular case of $\frac
12$-Laplacian, where the order $\frac 12$ is known as the critical order in
this context. More importantly, these equations are not translation invariant
and we prove that the viscosity solutions of such equations are $C^{1,\sigma}$,
making the equations classically solvable.
