The population balances describing the time dependence of the size distribution can, under some conditions, be trasformed by means of a similarity transformation into an ordinary integro-differential equation containing two instead of three variables. If there is compatibility between the transformed equation and the constraints given by the total mass conservation equation and the equation for the total number of particles, a self-preserving spectrum of the first kind can be obtaiened. There are, however, many situations such as the sintering controlled aging of supported metal catalysts, coagulation of colloidal particles in laminar shear flow, and coagulation of colloidal particles in a turbulent flow when the particles are smaller than the size of the smallest eddy for which, although a similarity transformation is possible, the transformed equation has no solution because of incompatibility with the above mentioned constraints. A second kind of self-preserving spectrum is suggested for these situations. The new variables are induced from a particular case for which an analytical result is available. A detailed presentation of the sintering controlled aging of supported metal catalysts is presented.