  A non-linear Penner type interaction is introduced and studied in the random
matrix model of homo-RNA. The asymptotics in length of the partition function
is discussed for small and large $N$ (size of matrix). The interaction doubles
the coupling ($v$) between the bases and the dependence of the combinatoric
factor on ($v,N$) is found. For small $N$, the effect of interaction changes
the power law exponents for the secondary and tertiary structures. The specific
heat shows different analytical behavior in the two regions of $N$, with a
peculiar double peak in its second derivative for N=1 at low temperature.
Tapping the model indicates the presence of multiple solutions.
