A scheme is proposed for analyzing bifurcations of equilibrium configurations of a weakly inhomogeneous elastic beam on an elastic foundation in the case of bimodal degeneration. For a homogeneous beam, a similar problem was solved by Darinskii and Sapronov in [1, 2]. Their analysis scheme is based on the condition that the eigenfunctions e 1 and e 2 of the second differential (at zero) of the energy functional are constant. In the case of an inhomogeneous beam, however, Darinskii and Sapronov’s analysis scheme has to be modified, since this condition is violated; moreover, it cannot be directly generalized, for example, in the form of an existence condition for a continuous family of eigenfunctions. We propose a scheme in which the constancy condition for eigenfunctions is replaced by the existence of a pair of smooth vector fields and whose linear span is invariant under the second differential at zero. Such a pair is sufficient for constructing the principal part of the key function and, as a consequence, for analyzing bifurcations of equilibrium configurations of the beam. The construction of the required pair of vector fields relies heavily on the formula from [3] for an orthogonal projector (onto the linear span of and ). In the case of a weakly inhomogeneous beam, the symmetry of the key function is reduced and the dimension of its strain basis is increased by one (a quadratic term is added to the principal part of the key function). In this situation, the analysis of bifurcation diagrams and bifurcating critical points is somewhat complicated [4] but reveals new bifurcation effects.