bifurcation problem with symmetry group

Let $\mathrm{\Gamma }$ be a Lie group acting on a vector space $V$ and let the system of ordinary differential equations

$$\dot{𝐱}+g(𝐱,\lambda )=0$$

where $g:{ℝ}^n\times ℝ\to {ℝ}^n$ is smooth. Then $g$ is called a bifurcation problem with symmetry group $\mathrm{\Gamma }$ if $g\in {\overrightarrow{ℰ}}_{x,\lambda }(\mathrm{\Gamma })$ (where $\overrightarrow{\mathrm{E}}\mathit{}\mathrm{(}\mathrm{\Gamma }\mathrm{)}$ is the space of $\mathrm{\Gamma }$-equivariant germs, at the origin, of $C^{\mathrm{\infty }}$ mappings of $V$ into $V$) satisfying

$$g(0,0)=0$$

and

$${(dg)}_{0,0}=0$$

where ${(dg)}_{0,0}$ denotes the Jacobian Matrix evaluated at $(0,0)$.[GSS]