For the parameter dependent nonlinear equation \(F(x,\lambda) =0\), \(F: \mathbb{R}^n \times \mathbb{R}^1 \to\mathbb{R}^n\), the generically important case \(\text{rank} \partial_x E(x^*, \lambda^*) =n-1\) is investigated. In a neighborhood of such pitchfork bifurcation point \((x^*, \lambda^*)\) of multiplicity \(p\geq 1\) the Lyapunov-Schmidt branching equation has the normal form \(g(\xi,\mu) =\pm \xi^{2+p} \pm \mu \xi=0\). It is shown that such points satisfy a minimally extended system \(G(y) =0\), \(G: \mathbb{R}^{n+2} \to \mathbb{R}^{n+2}\), where the dimension \(n+2\) is independent of \(p\). For solving this system a two-stage Newton-type method is suggested. The method is illustrated by numerical experiments.