We propose a method for the design of metalenses generating and focusing so-called vector Lissajous beams (VLBs), a generalization of cylindrical vector beams (CVBs) in the form of vector beams whose polarization vector is defined by two orders (p, q). The designed metalenses consist of subwavelength gratings performing the polarization transformation of the incident linearly polarized laser beams and a sublinearly chirped lens term for the realization of the beam focusing. The possibility of using VLBs for the realization of laser beams with a complex Poynting vector is theoretically shown. The certain choice of orders (p, q) of the generated VLBs makes it possible to control the type of various electromagnetic field components as well as the components of the complex Poynting vector. For example, in contrast to VLBs, the classical types of CVBs cannot provide an imaginary part in the longitudinal component of the Poynting vector. Such light fields are promising for exciting non-standard forces acting on the trapped nano- and microparticles.