A parameterized theory of convection is developed for 6 medium-size icy satellites (MIS) of Saturn. It is an extension of the research concerning the Mimas-Enceladus paradox. Two parameterizations of dimensionless temperature are used in the model and a new constrain for tidal heating is included. It is found that the basic results of the model are independent of particulars of the parameterizations. The new constrain considerably reduces the space of possible values of the material parameter of satellites but the two basic conclusions are unchanged, i.e.: (a) the thermal state of the considered MIS can be explained in the frame of the uniform model that includes radiogenic and tidal heating; (b) the theory indicates that endogenic activity of some MIS was (or is) a result of a specific ‘excited’, high temperature state of a given satellite. The theory could be also used for estimation of tidal heating.