  It is known that the classical limit of the second order BPZ null vector
decoupling equation for the simplest two 5-point degenerate spherical conformal
blocks yields: (i) the normal form of the Heun equation with the complex
accessory parameter determined by the 4-point classical block on the sphere,
and (ii) a pair of the Floquet type linearly independent solutions. A key point
in a derivation of the above result is the classical asymptotic of the 5-point
degenerate blocks in which the so-called heavy and light contributions
decouple. In the present work the semi-classical heavy-light factorization of
the 5-point degenerate conformal blocks is studied. In particular, a mechanism
responsible for the decoupling of the heavy and light contributions is
identified. Moreover, it is shown that the factorization property yields a
practical method of computation of the Floquet type Heun's solutions. Finally,
it should be stressed that tools analyzed in this work have a broad spectrum of
applications, in particular, in the studies of spectral problems with the Heun
class of potentials, sphere-torus correspondence in 2d CFT, the KdV theory, the
connection problem for the Heun equation and black hole physics. These
applications are main motivations for the present work.
