  This paper studies the dynamics and integrability of a variable-length
coupled pendulum system. The complexity of the model is presented by joining
various numerical methods, such as the Poincar\'e cross-sections,
phase-parametric diagrams, and Lyapunov exponents spectra. We show that the
presented model is hyperchaotic, which ensures its nonintegrability. We gave
analytical proof of this fact analyzing properties of the differential Galois
group of variational equations along certain particular solutions of the
system. We employ the Kovacic algorithm and its extension to dimension four to
analyze the differential Galois group. Amazingly enough, in the absence of the
gravitational potential and for certain values of the parameters, the system
can exhibit chaotic, integrable, as well as superintegrable dynamics. To the
best of our knowledge, this is the first attempt to use the method of Lyapunov
exponents in the systematic search for the first integrals of the system. We
show how to effectively apply the Lyapunov exponents as an indicator of
integrable dynamics. The explicit forms of integrable and superintegrable
systems are given.
  The article has been published in Nonlinear Dynamics, and the final version
is available at this link: https://doi.org/10.1007/s11071-023-09253-5
