  For a reaction network with species set $\mathscr{S}$, a log-parametrized
(LP) set is a non-empty set of the form $E(P, x^*) = \{x \in
\mathbb{R}^\mathscr{S}_> \mid \log x - \log x^* \in P^\perp\}$ where $P$
(called the LP set's flux subspace) is a subspace of $\mathbb{R}^\mathscr{S}$,
$x^*$ (called the LP set's reference point) is a given element of
$\mathbb{R}^\mathscr{S}_>$, and $P^\perp$ (called the LP set's parameter
subspace) is the orthogonal complement of $P$. A network with kinetics $K$ is a
positive equilibria LP (PLP) system if its set of positive equilibria is an LP
set. Analogously, it is a complex balanced equilibria LP (CLP) system if its
set of complex balanced equilibria is an LP set. An LP kinetic system is a PLP
or CLP system. This paper studies concentration robustness of a species on
subsets of equilibria. We present the "species hyperplane criterion", a
necessary and sufficient condition for absolute concentration robustness (ACR)
for a species of a PLP system. An analogous criterion holds for balanced
concentration robustness (BCR) for species of a CLP system. These criteria also
lead to interesting necessary properties of LP systems with concentration
robustness. Furthermore, we show that PLP and CLP power law systems with
Shinar-Feinberg reaction pairs in species $X$ in a linkage class have ACR and
BCR in $X$, respectively. This leads to a broadening of the "low deficiency
building blocks" framework to include LP systems of Shinar-Feinberg type with
arbitrary deficiency. Finally, we apply our results to species concentration
robustness in LP systems with poly-PL kinetics.
