  In this work, we continue the development of methods for constructing Lax
pairs and recursion operators for nonlinear integrable hyperbolic equations of
soliton type, previously proposed in the work of Habibullin et al. (2016 {\it
J. Phys. A: Math. Theor.} {\bf 57} 015203). This approach is based on the use
of the well-known theory of Laplace transforms. The article completes the proof
that for any known integrable equation of sine-Gordon type, the sequence of
Laplace transforms associated with its linearization admits a third-order
finite-field reduction. It is shown that the found reductions are closely
related to the Lax pair and recursion operators for both characteristic
directions of the given hyperbolic equation. Previously unknown Lax pairs and
recursion operators were constructed.
