  Solving tangle equations is deeply connected with studying enzyme action on
DNA. The main goal of this paper is to solve the system of tangle equations
$N(O+X_1)=b_1$ and $N(O+X_2)=b_2 \# b_3$, where $X_1$ and $X_2$ are rational
tangles, and $b_i$ is a 2-bridge link, for $i=1,2,3$, with $b_2$ and $b_3$
nontrivial. We solve this system of equations under the assumption
$\widetilde{O}$, the double branched cover of $O$, is not hyperbolic, i.e.$O$
is not $\pi$-hyperbolic. Besides, we also deal with tangle equations involving
2-bridge links only under the assumption $O$ is an algebraic tangle.
