In this work, we propose a synthetical integrable two-component model with peakon and weak kink solutions. The two-component model is proven integrable through its Lax pair and infinitely many conservation laws. Moreover, we investigate the bi-Hamiltonian structure and the interaction of multi-peakons from the two-component model. In the paper, we also study some reductions from the two-component model and obtain very interesting solutions, such as new type of N-peakon solutions, N-weak-kink solutions, weak hat-shape soliton solutions, periodic-peakon solutions, and rational peakon solutions. In particular, we first time propose N-weak-kink solutions, rational peakon solutions, and weak hat-shape soliton solutions.