  A new method to find first integrals of nonlinear differential equations in
Jacobi-type form is presented. The basic idea of our approach is to use
one-parameter perturbed motions to find well-conceived nonlocal constants that
are conserved along solutions. By means of such nonlocal framework we derive a
set of theorems that we apply to look for the first integrals of some relevant
cases, where moreover a solution is obtained. Applications also include some
equations of the Painlev\'{e}-Gambier classification.
