  Surface plasmon spectrum of a metallic hyperbola can be found analytically
with the separation of variables in elliptic coordinates. The spectrum consists
of two branches: symmetric, low-frequency branch, $\omega <\omega_0/\sqrt{2}$,
and antisymmetric high-frequency branch, $\omega >\omega_0/\sqrt{2}$, where
$\omega_0$ is the bulk plasmon frequency. The frequency width of the plasmon
band increases with decreasing the angle between the asymptotes of the
hyperbola. For the simplest multi-connected geometry of two hyperbolas
separated by an air spacer the plasmon spectrum contains two low-frequency
branches and two high-frequency branches. Most remarkably, the lower of two
low-frequency branches exists at $\omega \rightarrow 0$, i.e., unlike a single
hyperbola, it is "thresholdless." We study how the complex structure of the
plasmon spectrum affects the energy transfer between two emitters located on
the surface of the same hyperbola and on the surfaces of different hyperbolas.
