The author is concerned with the problem of looking for \(H\)-distinguished automorphic representations of \(G\), where \(G\) is a reductive algebraic group and \(H\) is a spherical subgroup of \(G\). By applying Arthur’s truncation method and the Rankin-Selberg integral representation method, the existence of \(H\)-distinguished residual representations of \(G\) and the relations to special values of automorphic \(L\)-functions have been obtained for the following three cases: \((G,H)= (D_4,G_2), (B_3,G_2)\), or \((G_2,A_2)\).