Two points lie in the same leaf if they are joined by the integral curve of a Hamiltonian vector field.

In other words, the graph of the solution must be a union of integral curves of this vector field.

These integral curves are called the characteristic curves of the original partial differential equation.

The integral curves of this vector field go from the points of the curve to the point P in finite time.

In this example, the other integral curves are all simple closed curves.

The map may be constructed as the integral curve of either the right- or left-invariant vector field associated with .

That the integral curve exists for all real parameters follows by right- or left-translating the solution near zero.

The map is the integral curve through the identity of both the right- and left-invariant vector fields associated to .

The integral curve through of the left-invariant vector field associated to is given by .

Likewise, the integral curve through of the right-invariant vector field is given by .