In this work, we use the popular Helfrich-Cahn-Hilliard phase field model for two-component lipid bilayer vesicles to systematically study phase transitions in lipid vesicles. We do this in the context of bifurcation theory. From a mathematical point-of-view, a particularly troubling aspect of lipid membrane behavior is its fluidity. This manifests as invariance of the energy functional under reparametrizations. The associated symmetry group in this case is the infinite-dimensional diffeomorphism group on S 2 . As a consequence, the EulerLagrange equations are underdetermined. By viewing this symmetry as an example of a gauge group, we propose a gauge fixing procedure using harmonic maps [13] to break the symmetry, thereby removing redundancies from the system. Applying standard tools of local bifurcation theory to the problem is not straightforward. The O(3) symmetry of the problem renders the linearized EulerLagrange equations with a degenerate null space. We use group theoretic strategies [53, 28] to tame this degeneracy and perform a local bifurcation analysis. We establish the existence of local symmetry-breaking branches of solutions bifurcating from a (trivial) spherical homogeneous state. We provide a few explicit examples of these branches in the case of octahedral and icosahedral symmetry using a technique developed by Poole [52]. We computationally study the system for axisymmetric solutions. The gauge fixed formulation is discretized using a Galerkin projection and detailed bifurcation diagrams are obtained using path-following by systematically exploring the parameter space. We efficiently compute stability of these branches, by employing the block-diagonalization technique using projective operator theory and assemble the hessian in its symmetry adapted basis. The eigen values of the individual blocks are used to assess stability. Explicit expression for the hessian is also presented. In literature, two possible formulations for the model are available that differ on the nature of the area constraint imposed. Here, we resolve this theoretical dilemma. We prove that lipid membranes of genus zero (homeomorphic to a sphere) can be modelled either as locally or globally area preserving by showing that the two formulations are equivalent with respect to determining equilibrium solutions and their stability.