  We investigate a generalisation of the three dimensional spring-pendulum
system. The problem depends on two real parameters $(k,a)$, where $k$ is the
Young modulus of the spring and $a$ describes the nonlinearity of elastic
forces. We show that this system is not integrable when $k\neq -a$. We
carefully investigated the case $k= -a$ when the necessary condition for
integrability given by the Morales-Ramis theory is satisfied. We discuss an
application of the higher order variational equations for proving the
non-integrability in this case.
