  The paper contains integral representations for certain classes of
exponentially growing solutions of second order periodic elliptic equations.
These representations are the analogs of those previously obtained by S. Agmon,
S. Helgason, and other authors for solutions of the Helmholtz equation. When
one restricts the class of solutions further, requiring their growth to be
polynomial, one arrives to Liouville type theorems, which describe the
structure and dimension of the spaces of such solutions. The Liouville type
theorems previously proved by M. Avellaneda and F.-H. Lin, and J. Moser and M.
Struwe for periodic second order elliptic equations in divergence form are
significantly extended. Relations of these theorems with the analytic structure
of the Fermi and Bloch surfaces are explained.
