  In this paper we solve the Bj\"orling problem for the class of immersed
surfaces in $\mathbb{R}^3$ whose mean curvature is given as an analytic
function depending on its Gauss map. As an application, we prove the existence
of surfaces with the topology of a M\"obius strip for an arbitrary large class
of prescribed functions. In particular, we use the Bj\"orling problem to
construct the first known examples of self-translating solitons of the mean
curvature flow with the topology of a M\"obius strip in $\mathbb{R}^3$
