  In order to solve a system of nonlinear rate equations one can try to use
some soliton methods. The procedure involves three steps: (1) Find a `Lax
representation' where all the kinetic variables are combined into a single
matrix $\rho$, all the kinetic constants are encoded in a matrix $H$; (2) find
a Darboux-Backund dressing transformation for the Lax representation $i\dot
\rho=[H,f(\rho)]$, where $f$ models a time-dependent environment; (3) find a
class of seed solutions $\rho=\rho[0]$ that lead, via a nontrivial chain of
dressings $\rho[0]\to \rho[1]\to \rho[2]\to\dots$ to new solutions, difficult
to find by other methods. The latter step is not a trivial one since a
non-soliton method has to be employed to find an appropriate initial $\rho[0]$.
Procedures that lead to a correct $\rho[0]$ have been discussed in the
literature only for a limited class of $H$ and $f$. Here, we develop a
formalism that works for practically any $H$, and any explicitly time-dependent
$f$. As a result, we are able to find exact solutions to a system of equations
describing an arbitrary number of species interacting through (auto)catalytic
feedbacks, with general time dependent parameters characterizing the
nonlinearity. Explicit examples involve up to 42 interacting species.
