Summary: This paper studies linear conjugacy of PL-RDK systems, which are kinetic systems with power law rate functions whose kinetic orders are identical for branching reactions, i.e. reactions with the same reactant complex. Mass action kinetics (MAK) systems are the best known examples of such systems with reactant-determined kinetic orders (RDK). We specify their kinetics with their rate vector and \(T\) matrix. The \(T\) matrix is formed from the kinetic order matrix by replacing the reactions with their reactant complexes as row indices (thus compressing identical rows of branching reactions of a reactant complex to one) and taking the transpose of the resulting matrix. The \(T\) matrix is hence the kinetic analogue of the network’s matrix of complexes \(Y\) with the latter’s columns of non-reactant complexes truncated away. For MAK systems, the \(T\) matrix and the truncated \(Y\) matrix are identical. We show that, on non-branching networks, a necessary condition for linear conjugacy of MAK systems and, more generally, of PL-FSK (power law factor span surjective kinetics) systems, i.e. those whose \(T\) matrix columns are pairwise different, is \(T = T'\), i.e. equality of their \(T\) matrices. This motivated our inclusion of the condition \(T = T'\) in exploring extension of results from MAK to PL-RDK systems. We extend the Johnston-Siegel criterion for linear conjugacy from MAK to PL-RDK systems satisfying the additional assumption of \(T = T'\) and adapt the MILP algorithms of M. D. Johnston et al. [J. Math. Chem. 50, No. 1, 274–288 (2012; Zbl 1238.92077)] and G. Szederkényi [J. Math. Chem. 47, No. 2, 551–568 (2010; Zbl 1198.92052)] to search for linear conjugates of such systems. We conclude by illustrating the results with several examples and an outlook on further research.