We study the the Euler-Lagrange equation of the dynamical Boulatov model, which is a simplicial model for 3D gravity augmented by a Laplace-Beltrami operator. We provide all its solutions on the space of left and right invariant functions that render the interaction of the model an equilateral tetrahedron. Surprisingly, for a nonlinear equation, the solution space forms a vector space. This space distinguishes three classes of solutions: saddle points, global and local minima of the action. Our analysis shows that there exists one parameter region of coupling constants, for which the action admits degenerate global minima.