This chapter explores the relationship between the existence of symmetries of a dynamical system (DS) and the presence of invariant manifolds under the dynamical flow. The focus is on center manifolds (CMs), which are crucial for analyzing DS dynamics. The system may depend on a real parameter λ, enabling the study of bifurcation problems. The approach extends known results from standard equivariant bifurcation theory to general Lie group symmetries. Classical results from Lyapunov-Schmidt projection are extended to CM reduction. The non-uniqueness of CMs leads to interesting features related to analyticity and perturbation expansions, discussed in terms of Poincaré-Dulac normal form reduction and the Shoshitaishvili theorem on topological equivalence of DS. The chapter begins with preliminary results on flow-invariant manifolds. For a DS described by $\dot{x} = f(x) = Ax + F(x)$ with a symmetry $s = s(x)$, the symmetry condition $\{f, s\} = [X_f, Y_s] = 0$ holds. A key lemma states that any invariant manifold under the dynamical flow is transformed by symmetries into another invariant manifold. The system is decomposed into subspaces $E_0$ and $E_1$, corresponding to eigenvalues with zero real part and other eigenvalues, respectively. The matrix A is put into block-diagonal form, and the system becomes $\dot{u} = A_0u + U(u,v)$, $\dot{v} = A_1v + V(u,v)$. A symmetry $Y_s = \varphi\partial_u + \psi\partial_v$ is considered, leading to results on the derivatives of $\psi_p$ and $\varphi_s$ at the origin. A lemma shows that transformations $Y_s$ preserving the symmetry conditions map manifolds tangent to v = 0 at the origin into other such manifolds. This highlights the role of symmetries in preserving the structure of invariant manifolds.This chapter explores the relationship between the existence of symmetries of a dynamical system (DS) and the presence of invariant manifolds under the dynamical flow. The focus is on center manifolds (CMs), which are crucial for analyzing DS dynamics. The system may depend on a real parameter λ, enabling the study of bifurcation problems. The approach extends known results from standard equivariant bifurcation theory to general Lie group symmetries. Classical results from Lyapunov-Schmidt projection are extended to CM reduction. The non-uniqueness of CMs leads to interesting features related to analyticity and perturbation expansions, discussed in terms of Poincaré-Dulac normal form reduction and the Shoshitaishvili theorem on topological equivalence of DS. The chapter begins with preliminary results on flow-invariant manifolds. For a DS described by $\dot{x} = f(x) = Ax + F(x)$ with a symmetry $s = s(x)$, the symmetry condition $\{f, s\} = [X_f, Y_s] = 0$ holds. A key lemma states that any invariant manifold under the dynamical flow is transformed by symmetries into another invariant manifold. The system is decomposed into subspaces $E_0$ and $E_1$, corresponding to eigenvalues with zero real part and other eigenvalues, respectively. The matrix A is put into block-diagonal form, and the system becomes $\dot{u} = A_0u + U(u,v)$, $\dot{v} = A_1v + V(u,v)$. A symmetry $Y_s = \varphi\partial_u + \psi\partial_v$ is considered, leading to results on the derivatives of $\psi_p$ and $\varphi_s$ at the origin. A lemma shows that transformations $Y_s$ preserving the symmetry conditions map manifolds tangent to v = 0 at the origin into other such manifolds. This highlights the role of symmetries in preserving the structure of invariant manifolds.