We prove that a H-surface M in H^2xR, H= 1/2 , inherits the symmetries of its boundary , when the boundary is either a horizontal curve with curvature greater than one or two parallel horizontal curves with curvature greater than one, whose distance is greater or equal to pi. Furthermore we prove that the asymptotic boundary of a surface with mean curvature bounded away from zero consists of parts of straight lines, provided it is sufficiently regular.