Bagpipes configurations in mechanics and electromagnetism

It is shown that when a first integral I of a vector field (v.f. in what follows) X is known, the level sets of I resemble bagpipes, and X is asymptotically stable (a.s. in what follows) on the skeleton of I (the set where ∇ I vanishes), then the v.f. is stable at 0 (a singular, not necessarily isolated, zero of X). A similar bagpipes configuration is shown to appear concerning the orbits of the magnetic field created by a set of concurrent straight line wires.