Let X be a smooth projective connected curve. Given a complex representation \(\rho \) of the exceptional simple complex Lie group \(E_6\) , an \((E_6,\rho )\) -Higgs pair over X is a pair \((E,\varphi )\) where E is a principal \(E_6\) -bundle over X and \(\varphi \) is a holomprphic global section of the vector bundle associated to E by \(\rho \) tensorized by the canonical line bundle over X. In this paper, \((E_6,\rho )\) -Higgs bundles are considered for the fundamental representations of \(E_6\) different from the adjoint representation. In particular, a result is proved that provides a specific vector form for these pairs and reduced notions of stability and polystability in terms of filtrations are also provided. The stability condition of the pairs is characterized for non-simple \((E_6,\rho )\) -Higgs pairs in relation to the mentioned vector form. Moreover, it is noted that, in addition to the adjoint representation, there is only one fundamental representation of \(E_6\) which is fixed by the action of the outer involution \(\sigma \) that \(E_6\) admits. In the paper, an action of \(\sigma \) on Higgs pairs with group \(E_6\) and associated to this specific representation is defined, and the fixed points of this action are computed. In particular, these fixed points are characterized in terms of the vector form of the pairs in the stable and non-simple case.
