We consider the equation ѱ(z+h)=M(z)ѱ(z), where z ∈ ℂ, h>0 is a parameter, and M:ℂ→SL(2,ℂ) is a given analytic function. We get asymptotics of its analytic solutions as h → 0. The asymptotic formulas contain an analog of the geometric (Berry) phase well known in the quasiclassical analysis of differential equations.