  Based on the technique of the discrete one-turn transfer maps, the problem of
linear coupling between horizontal and vertical betatron oscillations in an
accelerator has been treated exactly and entirely in explicit form. The
stability region in the fractional part of the horizontal and the vertical
betatron tune space as a function of the linear coupling strength, has been
obtained, and the increment/decrement of the horizontal and the vertical
betatron oscillations in the case of the linear sum resonance has been shown to
be approximately equal to the half of the coupling strength.
  The normal form parameterization of the one-turn linear map with
horizontal-to-vertical coupling has been developed in detail in the spirit of
the Edwards and Teng formalism. The motion in the normal mode in the new normal
form coordinates is decoupled implying that two independent Courant-Snyder
invariants exist, which have been found explicitly.
