We prove the existence of integral (stable, unstable, center) manifolds of admissible classes for the solutions to the semilinear integral equation $u\left(t\right)=U(t,s)u\left(s\right)+{\int }_s^{t}U(t,\xi )f(\xi ,u\left(\xi \right))d\xi$ when the evolution family ${\left(U(t,s)\right)}_{t\ge s}$ has an exponential trichotomy on a half-line or on the whole line, and the nonlinear forcing term f satisfies the (local or global) φ-Lipschitz conditions, i.e., ||f(t,x)-f(t,y)|| ≤ φ(t)||x-y|| where φ(t) belongs to some classes of admissible function spaces. These manifolds are formed by trajectories of the solutions belonging...