A generalization for the physically nonlinear Maxwell-type constitutive equation is proposed with two material functions for non-aging rheonomic materials, which have been studied analytically in previous articles to elucidate its properties and application. To extend the set of basic rheological phenomena that it simulates, we propose to add the third strain component expressed as the Boltzmann-Volterra linear integral operator governed by an arbitrary creep function. To generalize and conveniently tune the constitutive relation, to fit it to various materials and various lists of phenomena (test data), we introduce a weighting factor (i.e. nonlinearity factor) into the equation. This allows us to crossbreed primary physically nonlinear Maxwell-type model with the linear viscoelasticity equation in an arbitrary proportion, to construct a hybrid model and to regulate prominence of different phenomena described by the two constitutive equations we crossbred. General expression for stress-strain curves at constant stress rate and for the creep and recovery curves families obtained using the proposed hybrid constitutive equation are derived and analyzed. The basic properties of the stress-strain curves and the creep-recovery curves are studied assuming three material functions are arbitrary. They are also compared to the properties obtained using primary Maxwell-type model and linear viscoelasticity theory. New properties are found that allow the hybrid model to tune the form of the stress-strain curves and the creep-recovery curves and to simulate additional effects observed in constant stress rate tests and creep-recovery tests of various materials at different stress rates and stress levels.