  A theorem is proved which determines the first integrals of the form
$I=K_{ab}(t,q)\dot{q}^{a}\dot{q}^{b}+K_{a}(t,q)\dot{q}^{a}+K(t,q)$ of
autonomous holonomic systems using only the collineations of the kinetic metric
which is defined by the kinetic energy or the Lagrangian of the system. It is
shown how these first integrals can be associated via the inverse Noether
theorem to a gauged weak Noether symmetry which admits the given first integral
as a Noether integral. It is shown also that the associated Noether symmetry is
possible to satisfy the conditions for a Hojman or a form-invariance symmetry
therefore the so-called non-Noetherian first integrals are gauged weak Noether
integrals. The application of the theorem requires a certain algorithm due to
the complexity of the special conditions involved. We demonstrate this
algorithm by a number of solved examples. We choose examples from published
works in order to show that our approach produces new first integrals not found
before with the standard methods.
