Thick brane solitons breaking $Z_2$ symmetry

New soliton solutions for thick branes in 4 + 1 dimensions are considered in this article. In particular, brane models based on the sine-Gordon (SG), ${\phi }^4$ and ${\phi }^6$ scalar fields are investigated; in some cases $Z_2$ symmetry is broken. Besides, these soliton solutions are responsible for supporting and stabilizing the thick branes. In these models, the origin of the symmetry breaking resides in the fact that the modified scalar field potential may have non-degenerate vacuua and these non-degenerate vacuua determine the cosmological constant on both sides of the brane. At last, in order to explore the particle motion in the neighborhood of the brane, the geodesic equations along the fifth dimension are studied.