For a rational function R, let \(N_R(z)=z-\frac{R(z)}{R'(z)}.\) Any such \(N_R\) is referred to as a Newton map. We determine all the rational functions R for which \(N_R\) has exactly two attracting fixed points, one of which is an exceptional point. Further, if all the repelling fixed points of any such Newton map are with multiplier 2, or the multiplier of the non-exceptional attracting fixed point is at most \(\frac{4}{5}\) , then its Julia set is shown to be connected. If a polynomial p has exactly two roots, is unicritical but not a monomial, or \(p(z)=z(z^n+a)\) for some \(a \in \mathbb {C}\) and \(n \ge 1\) , then we have proved that the Julia set of \(N_{1/p}\) is totally disconnected. For the McMullen map \(f_{\lambda }(z)=z^m - \frac{\lambda }{z^n}\) , \(\lambda \in \mathbb {C}{\setminus } \{0\}\) and \(m,n \ge 1\) , we have proved that the Julia set of \(N_{f_\lambda }\) is connected and is invariant under rotations about the origin of order \(m+n\) . All the connected Julia sets mentioned above are found to be locally connected.
