A family of exact relativistic stellar models is described. The family generalizes Buchdahl's n = 1 polytropic solution. The matter content is a perfect fluid and, excluding Buchdahl's original model, it behaves as a liquid at low pressures in the sense that the energy density is non-zero in the zero-pressure limit. The equation of state has two free parameters, a scaling and a stiffness parameter. Depending on the value of the stiffness parameter the fluid behaviour can be divided into four different types. Physical quantities such as masses, radii and surface redshifts as well as density and pressure profiles are calculated and displayed graphically. Leaving the details to a later publication, it is noted that one of the equation of state types can quite accurately approximate the equation of state of real cold matter in the outer regions of neutron stars. Finally, it is observed that the given equation of state does not admit models with a conical singularity at the centre.