A theorem is proved that determines the first integrals of the form I=Kab(t,q)q̇aq̇b+Ka(t,q)q̇a+K(t,q) of autonomous holonomic systems using only the collineations of the kinetic metric that 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 with 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.

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Equivalently, we may assume Kab=N=1nfN(t)D(N)ab(q), where fN(t) is a sequence of analytic functions and D(N)ab is a sequence of KTs of γab. This expression is equivalent to (37) because if we write
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28.

We note that the FI J1 is for n finite, whereas J2 is for n infinite, hence the term eλt.

29.

We note that for n = 0, the conditions for the QFI J1(n = 0) can be derived if we set equal to zero the quantities C(N)ab and L(N)a for N≠0.

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If, in addition, Fa = 0 the Qa = V,a the case is reduced to that of the autonomous conservative systems.

31.

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43.

Note that La in (54) is the sum of the non-proper ACs of E2 and not of its KVs which give Cab = 0.

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46.

We ignore the index (0) in order to simplify the notation.

47.

If we set z = it, the line element (60) takes the form ds2=dt2t2dx2+dy2, which is a conformally flat spacetime.

48.

For simplicity, we set C(0)abCab, L(0)aBa, C(1)abDab and L(1)aLa.

49.

Equation (43) is not necessary because the integrability condition K,[ab] = 0 does not intervene in the calculations. However, it has been checked that Eq. (43) is always satisfied identically from the solutions of the other equations of the system.

50.

To find a solution, we consider C(0)ab = c0Cab, C(1)ab = c1Cab, …, C(n)ab = ncnCab, L(0)a = d0La, L(1)a = d1La, …, L(m)a = dmLa.

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