  A class of autonomous, even-order ordinary differential equations is
discussed from the point of view of Lie symmetries. It is shown that for a
certain power nonlinearity, the Noether symmetry group coincides with the Lie
point symmetry group. First integrals are established and exact solutions are
found. Furthermore, this paper complements, for the one-dimensional case, some
results in the literature of Lie group analysis of poliharmonic equations and
Noether symmetries obtained in the last twenty years. In particular, it is
shown that the exceptional negative power discovered in [A. H. Bokhari, F. M.
Mahomed and F. D. Zaman, Symmetries and integrability of a fourth-order
Euler-Bernoulli beam equation, J. Math. Phys., vol. 51, 053517, (2010)] is a
member of an one-parameter family of exceptional powers in which the Lie
symmetry group coincides with the Noether symmetry group.
