The authors prove new bifurcation existence theorems based on the theory of second-order optimality conditions for abnormal constrained optimization problems [the first author, Optimality conditions. Abnormal and degenerate problems. Dordrecht: Kluwer Academic Publishers (2000; Zbl 0987.49001)]. Remarks: In the list of references the following articles should be included: \(1^0\) \textit{V. A. Trenogin} and \textit{N. A. Sidorov} [An investigation of the bifurcation points and nontrivial branches of the solutions of nonlinear equations. (Russian), Irkutsk., Gos. Univ., Irkutsk: Differential and integral equations, No. 1, 216-247 (1972)], where the most general theorem on the bifurcation near an eigenvalue of an odd multiplicity was proved. \(2^0\) \textit{V. A. Trenogin, N. A. Sidorov} and \textit{B. V. Loginov} [Differ. Integral Equ. 3, No. 1, 145-154 (1990; Zbl 0729.47060)]. \(3^0\) \textit{V. A. Trenogin} and \textit{N. A. Sidorov} [Potentiality conditions for a branching equation and bifurcation points of nonlinear operators. (Russian), Uzbek. Mat. Zh., No. 2, 40-49 (1992)] where general theorems on the bifurcation existence under branching equation potentiality conditions have been proved.