This chapter focuses on the relationship between symmetries of a dynamical system (DS) and the existence of invariant manifolds, particularly center manifolds (CM). The study assumes that the system's dynamics may depend on a real parameter, allowing for the consideration of bifurcation problems. The approach extends known results from standard equivariant bifurcation theory to the case of general linearly parameterized (LP) symmetries. Key aspects include the non-uniqueness of CM and its implications for analyticity and perturbation expansions, as well as the application of the Poincaré-Dulac normal form reduction and the Shoshitaishvili theorem on topological equivalence of DSs. The chapter begins with preliminary results on flow-invariant manifolds. It introduces the concept of a symmetry \( Y_s \) and demonstrates that any manifold invariant under the dynamical flow is transformed by \( Y_s \) into another flow-invariant manifold. This is proven using the symmetry condition \( \{f, s\} = [X_f, Y_s] = 0 \). The chapter then discusses the block-diagonal form of the matrix \( A \) and the corresponding linear change of coordinates. It presents Lemmas 2 and 3, which provide conditions for the symmetry \( Y_s \) to preserve the structure of the DS. Specifically, Lemma 2 shows that certain partial derivatives of the symmetry components must be zero, and Lemma 3 ensures that transformations generated by \( Y_s \) preserve the smoothness and tangency of manifolds tangent to the subspace \( v = 0 \).This chapter focuses on the relationship between symmetries of a dynamical system (DS) and the existence of invariant manifolds, particularly center manifolds (CM). The study assumes that the system's dynamics may depend on a real parameter, allowing for the consideration of bifurcation problems. The approach extends known results from standard equivariant bifurcation theory to the case of general linearly parameterized (LP) symmetries. Key aspects include the non-uniqueness of CM and its implications for analyticity and perturbation expansions, as well as the application of the Poincaré-Dulac normal form reduction and the Shoshitaishvili theorem on topological equivalence of DSs. The chapter begins with preliminary results on flow-invariant manifolds. It introduces the concept of a symmetry \( Y_s \) and demonstrates that any manifold invariant under the dynamical flow is transformed by \( Y_s \) into another flow-invariant manifold. This is proven using the symmetry condition \( \{f, s\} = [X_f, Y_s] = 0 \). The chapter then discusses the block-diagonal form of the matrix \( A \) and the corresponding linear change of coordinates. It presents Lemmas 2 and 3, which provide conditions for the symmetry \( Y_s \) to preserve the structure of the DS. Specifically, Lemma 2 shows that certain partial derivatives of the symmetry components must be zero, and Lemma 3 ensures that transformations generated by \( Y_s \) preserve the smoothness and tangency of manifolds tangent to the subspace \( v = 0 \).