Consider an electric dipole and its electric field lines.

There will be many field lines that do not extend to or originate from infinity, but rather begin at the positive charge and loop back around to terminate at the negative charge.

However, the field lines exactly on the dipole axis will extend out to infinity (or originate from infinity).

But are these the only field lines that extend to infinity, or are there other off-axial lines that also go to infinity? How can one prove this?

I have tried writing out the differential equations for the field lines, but they seem impossible to solve. No heuristic argument has satisfied me either. One can assert that all off-axial electric field lines must curve away from the axis as one goes to infinity, but does this curvature ultimately cause them to curve all the way back or do they reach some sort of linear asymptote?