On existence of an x-integral for a semi-discrete chain of hyperbolic   type
  A class of semi-discrete chains of the form $t_{1x} = f(x, t, t_1, t_x)$ is considered. For the given chains easily verifiable conditions for existence of x-integral of minimal order 4 are obtained. 
Kostyantyn Zheltukhin and Natalya Zheltukhina
nlin.SI