The article is devoted to the improvement of the vibration isolation and shockproof properties of the shock absorber used in the vibration isolation compensator system of the pumping unit (PU). The set task is to create an elastic damping system with the desired low (quasi-zero) stiffness, which makes it possible to reduce the detrimental effect of pumping unit vibrations both on human health and on the equipment itself. High internal dynamic (vibration) loads are the main causes of early failure of pumping units, which are transmitted to the equipment through pipelines and foundations, mainly due to various operational factors. A promising trend in increasing the operating reliability and efficiency of pumping and power equipment is the use of a vibration-isolating compensator system (VICS). To obtain greater efficiency of vibration isolation, it is proposed to apply a modern method of vibration damping – the use of elastic mechanical systems with quasi-zero stiffness. The damping devices with a quasi-zero stiffness effect were studied. An assessment of the operability and effectiveness of these dampers at oil and gas facilities was given. Based on the information studied, a new concept of a quasi-zero stiffness damping device is proposed. The method of mathematical analysis determined its power curve. A 3D model of the damper device was built using a simulation software. The functional check of the damper was performed with mathematical modeling carried out on the computing device with the help of a specialized Ansys software that allows performing a system analysis of an object using the finite element method. The following results were obtained: the design of the damper with the effect of quasi-zero stiffness allows to widen the range of basement load transmission coefficient reduction towards low frequencies. The low rigidity of the system at the operating point ensures low values of the frequency of natural oscillations, and, consequently, high vibration isola