The paper is concerned with a transversely isotropic homogeneous elastic medium subjected to uniform compression in the isotropy plane. The medium becomes unstable in the sense of Hadamard [1] at a definite level of initial strain. The critical strain is established to be uniquely determinate from the system of equations of bifurcation of equilibrium; however, there are many modes of buckling corresponding to this strain. A solution of the system of equations of bifurcation is built in the form of doubly periodic functions sinr ₁x ₁sinr ₂x ₂. The uncertainty of the mode of buckling consists in the fact that the wave numbers r ₁ and r ₂ remain arbitrary. In order to determine the relationship between the wave numbers we examine the initial supercritical behavior of the material. It turns out that the only possible modes are the chess-board mode (with r ₁ = r ₂) and the corrugation-type mode (when one of the wave numbers r ₁ or r ₂ vanishes). The initial supercritical equilibrium is shown as being stable.