  We study the U(2) lattice gauge theory in the pure gauge sector using the
simplest action, with determinant and fundamental terms, having the naive
continuum limit of SU(2)$\times$U(1). We determine part of the phase diagram of
the model and find a first-order critical line which goes through the U(1)
critical point. We show how to deduce both the order parameter of the
first-order transition and the U(2) renormalization group flow from the lattice
potential in the determinant and fundamental representations. We give evidence
that a Monte-Carlo simulation of the model is indeed consistent with the
symmetric SU(2)$\times$U(1) continuum limit in the weak coupling pertubative
regime.
