For dilute telechelic linear and ring Rouse chains undergoing reversible end-association and dissociation, the time (t) evolution equation was analytically formulated for the bond vector of the subchain (or segment), u([c])(n,t) with n being the subchain index and the superscript c specifying the chain (c = L and R for the linear and ring chains). The end-association of the linear chain (i.e., ring formation) occurs only when the ends of the linear chain come into close proximity. Because of this constraint for the ring formation, the time evolution equation for u([L])(n,t) of the linear chain was formulated with a conceptually new, two-step expansion method: u([L])(n,t) was first expanded with respect to its sinusoidal Rouse eigenfunction, sin(p pi n/N) with p = integer and N being the number of subchains per chain, and then the series of odd sine modes is re-expanded with respect to cosine eigenfunctions of the ring chain, cos(2 alpha pi n/N) with alpha = integer, so as to account for that constraint. This formulation allowed analytical calculation of the orientational correlation function, S-[c](n,m,t) = b(-2)< u(x)([c])(n,t)u(y)([c])(m,t)> (c = L, R) with b being the subchain step length, and the viscoelastic relaxation function, g([c])(t) proportional to integral S-N(0)[c](n,n,t) dn. It turned out that the terminal relaxation of g([R])(t) and g([L])(t) of the ring and linear chains is retarded and accelerated, respectively, due to the motional coupling of those chains occurring through the reaction. This coupling breaks the ring symmetry (equivalence of all subchains of the ring chain in the absence of reaction), thereby leading to oscillation of the orientational anisotropy S-[R](n,n,t) of the ring chain at long t with the subchain index n. The coupling also reduces a difference of the anisotropy S-[L](n,n,t) of the linear chain at the middle (n similar to N/2) and end (n similar to 0).