Is Nöther Symmetric Approach Consistent With Dynamical Equation In Non-Minimal Scalar-Tensor Theories?

The form of the coupling of the scalar field with gravity and the potential have been found by applying N öther theorem to 2-dimensional minisuperspaces in induced gravity model. It has been observed that though the forms thus obtained are consistent with all the equations £XL = 0 , yet they do not satisfy the field equations fork = ±1 , in Robertson-Walker model. It has been pointed out that this is not due to the degeneracy of the Lagrangian, since this problem does not appear in k = 0 case. It has also been shown that though N öther theorem fails to extract any symmetry from the Lagrangian of such model for k = ±1, symmetry exists, which can easily be found by studying the continuity equation.