The authors consider a nonlinear equation \(f(x,t)=0,\) where \(x \in \mathbb R^{n}\) is a decision variable, \(t \in \mathbb R^{m}\) is a parameter, and \(f : \mathbb R^{n} \times \mathbb R^{m} \rightarrow \mathbb R^{n}\) is a smooth function and investigate monotonicity issues of the solution trajectory (path) \(x(t)\) (or some function \(\phi (x)\)) with respect to the parameter \(t\). Extending the results from the previous paper [J. Optim. Theory Appl. 137, No. 3, 641--673 (2008; Zbl 1229.90219)], they give conditions which provide existence of partial or full monotonicity paths, suggest a suitable selection method, and reveal some quantitative properties. Applications to equality constrained parametric optimization, optimal consumption, special two-player games, productivity estimation, and comparative dynamics are described.