We prove in this article some general steady state bifurcation theorem
for a class of nonlinear eigenvalue problems, in the case where algebraic
multiplicity of the eigenvalues of the linearized problem is even. These
theorems provide an addition to the classical Krasnoselskii and Rabinowitz
bifurcation theorems, which require the algebraic multiplicity of the
eigenvalues is odd. For this purpose, we prove a spectral theorem for
completely continuous fields, which can be considered as a generalized
version of the classical Jordan matrix theorem and the Fredholm theorem for
compact operators. An application to a system of second order elliptic
equations is given as well.