The paper deals with a two-parameter system of ordinary differential equations on \(\mathbb{R}^ n\) \((1) : \dot x = f(x, \lambda, \alpha)\), where \(f\) is \(Z_ 2\)-symmetric, i.e. there exists a linear mapping \(g : \mathbb{R}^ n \to \mathbb{R}^ n\) such that \(g^ 2 = I\), \(g \neq I\) and \(gf(x, \lambda, \alpha) = f(gx, \lambda, \alpha)\) for all \((x, \lambda, \alpha) \in \mathbb{R}^ n \times \mathbb{R}^ 2\). Let us denote \(X_ s = \{x \in \mathbb{R}^ n, gx = x\}\), \(X_ a = \{x \in \mathbb{R}^ n, gx = -x\}\). Further we assume \(f(0,0,0) = 0\) and the Jacobi matrix \(f_ x(0,0,0)\) has double zero eigenvalue such that the corresponding eigenspace is spanned by the vectors \(v_ s\), \(v_ a\) for which \(v_ s \in X_ s\) and \(v_ a \in X_ a\). Under certain nondegeneracy conditions the authors prove that for any \((\lambda, \alpha)\) near (0,0) with \(\alpha > 0\), the equation (1) possesses a pair of hyperbolic saddle points \(S_ +(\lambda, \alpha)\), \(S_ - (\lambda, \alpha)\) connected by a heteroclinic orbit \(\Gamma_ 0(\lambda, \alpha) \subset X_ s\) and in the \(\lambda - \alpha\) plane, there exists a curve \(C\) emanating from (0,0) such that for any \((\lambda, \alpha) \in C\), the equation (1) has two heteroclinic orbits \(\Gamma_ +\) and \(\Gamma_ -\) connecting \(S_ +\) and \(S_ -\), where \(\Gamma_ +\), \(\Gamma_ -\) are not in the symmetric subspace \(X_ s\) but are symmetric about \(X_ s\). The proof is based on the reduction of (1) into a center manifold, which inherits the \(Z_ 2\)- symmetry of (1) and using an appropriate normal form on this center manifold.