  Transfer systems are combinatorial objects which classify $N_\infty$ operads
up to homotopy. By results of A. Blumberg and M. Hill, every transfer system
associated to a linear isometries operad is also saturated (closed under a
particular two-out-of-three property). We investigate saturated and linear
isometric transfer systems with equivariance group $C_{p^mq^n}$, the cyclic
group of order $p^mq^n$ for $p,q$ distinct primes and $m,n\ge 0$. We give a
complete enumeration of saturated transfer systems for $C_{p^mq^n}$. We also
prove J. Rubin's saturation conjecture for $C_{pq^n}$; this says that every
saturated transfer system is realized by a linear isometries operad for $p,q$
sufficiently large (greater than $3$ in this case).
