We continue the study of additive functions fk:R→F(1≤k≤n) linked by an equation of the form ∑k=1npk(x)fk(qk(x))=0, where the pk and qk are polynomials, R is an integral domain of characteristic 0, and F is the fraction field of R. A method is presented for solving all such equations. We also consider the special case ∑k=1nxmkfk(xjk)=0 in which the pk and qk are monomials. In this case we show that if there is no duplication, i.e. if (mk,jk)≠(mp,jp) for k≠p, then each fk is the sum of a linear function and a derivation of order at most n-1. Furthermore, if this functional equation is not homogeneous then the maximal orders of the derivations are reduced in a specified way. [ABSTRACT FROM AUTHOR]