The author considers a one-parameter family of smooth vector fields
\[ du/dt=V(u,\mu), \quad u\in \mathbb R^m,\;\mu\in [-\mu_0,\mu_0],\;\mu_0>0, \]
such that the origin \(O\) is a fixed point, i.e., \(V(0,\mu)=0\), and assume the family is reversible with respect to two symmetries, i.e., there are \(S_j\in GL_m(\mathbb R), j=1,2\) such that \(S_j^2=I\) and \(S_jV(u,\mu)=-V(S_j u,\mu)\) for all \(u, \mu\) and for \(j=1,2\). Two cases are discussed:

(i)

\(O\) is a \(0^2\) resonant fixed point (i.e., \(L:=D_u V(0,0)\) has a double non-semi-simple eigenvalue \(0\) and no other eigenvalues with zero real part);

(ii)

\(O\) is a \(0^2i\omega\) resonant fixed point (i.e., \(L\) has two simple eigenvalues \(\pm i\omega\) with \(\omega>0\), a double non-semi-simple eigenvalue \(0\) and no other eigenvalues with zero real part).

Moreover, he assumes \(A_1:=\langle D^2_{\mu,u}V(0,0)\varphi_0, \varphi_1^* \rangle \neq 0\), i.e., not considering the case that the double eigenvalue \(0\) stay at \(0\) for \(\mu\neq 0\). Under these hypotheses, the author investigates the existence of homoclinic and heteroclinic orbits. For the \( 0^2\) resonance he proves the existence of homoclinic connections to \(O\) and of heteroclinic orbits. For the \( 0^2i\omega\) resonance he proves that in most of the cases the second symmetry induces the existence of homoclinic connections to \(O\) and of heteroclinic orbits whereas with a unique symmetry there is generically no homoclinic connection to \(O\). His main idea is: Apply the center manifold reduction to obtain a 2-dimensional system; then, approximate the reduced system by the normal form theorem with an integrable vector field; finally, use the persistence of reversible homoclinic or heteroclinic orbits for perturbed reversible vector fields in \(\mathbb R^2\) to prove the persistence of non-persistence of the homoclinic (or heteroclinic) solutions.