  We study the spontaneous temporal symmetry breaking instability in a
coherently-driven passive optical Kerr resonator observed experimentally by Xu
and Coen in Opt.~Lett.~{\bf 39}, 3492 (2014). We perform a detailed stability
analysis of the Lugiato-Lefever model for the optical Kerr resonators and
analyze the temporal bifurcation structure of stationary symmetric and the
emerging asymmetric states as a function of the pump power. For intermediate
pump powers a pitchfork loop is responsible for the destabilization of
symmetric states towards stationary asymmetric ones while at large pump powers
we find the emergence of periodic asymmetric solutions via a Hopf bifurcation.
From a theoretical perspective, we use local bifurcation theory in order to
analyze the most unstable eigenmode of the system. We also explore a
non-conservative variational approximation capturing, among others, the
evolution of the solution's amplitude, width and center of mass. Both methods
provide insight towards the pitchfork bifurcations associated with the symmetry
breaking.
