Based on the Lie theoretical methods of algebraic Fourier transformation, we classify in the case of generic values of inducing parameters the scalar singular vectors corresponding to the diagonal branching rules for scalar generalized Verma modules in the case of orthogonal Lie algebra and its conformal parabolic subalgebra with commutative nilradical, thereby realizing the diagonal branching rules in an explicit way. The complicated combinatorial structure of singular vectors is conveniently determined in terms of recursion relations for the generalized hypergeometric function \({ }_3F_2\) . As a geometrical application, we classify bilinear conformally equivariant differential operators acting on homogeneous line bundles on the flag manifold given by conformal sphere \(S^n\) .