In this paper, we investigate the continuation of an imbedded solution $x(t) = x(t,\alpha )$ of \[x(t) = f(t) + \int_0^\alpha {k(t,s)x(s)ds = f(t) + (K_\alpha x)(t)} \] through its first “critical point” $\alpha = c$. Under the assumption that the Fredholm resolvent $\Gamma _\alpha = (I - K_\alpha )^{ - 1} - I$ has a simple pole in its meromorphic expansion about $\alpha = c$ we obtain a simple eigenspace corresponding to$\lambda = 1$ for the operator $K_c $ ; and in accordance with the redholm alter-native, we have an imbedded solution for $\alpha > c$ for forcing functions orthogonal to the one-dimensional eigenspace of the adjoint operator $K_c^ * $. The principal technique is the explicit solving of the Bartle-Schmidt bifurcation equation.