An implant stabilizes two adjacent bones of a joint, while enabling a natural kinematic relative movement of the bones. Support components are connected to each bone of the joint, and a flexible core is interposed between them. The core and at least one of the support components are provided with a smooth sliding surface upon which the core and support component may slide relative to each other, enabling a corresponding movement of the bones. The surfaces may have a mating curvature, to mimic a natural movement of the joint. The core is resilient, and may bend or compress, enabling the bones to move towards each other, and or to bend relative to each other.