when the above expressions are properly defined. From where, we obtain again deed, the above formula yields Fig. 2. Sign of the polynomial P;, .(z,k1,k2) in K. 5.5. Integrable TDHOs and t-dependent constants of the motion. The autonomisations of the \ 92 7 NBL ey 8 v2 T 2 ¥ J \ §2 7 Only the solutions of the above system obeying that x7 + x3 + y?7 + y} = 1 describe curve: in SU(2) and, consequently, are related to solutions of system (6.7). Nevertheless, we can forge such a restriction for the time being, because it can be automatically implemented later in a more suitable way. Therefore, we can deal with the four variables in the preceding system of differentia equations (6.8) as if they were independent. This linear system of differential equations is < Lie system associated with a Lie algebra of vector fields gl(4, IR), but the solutions with initia condition related to a matrix in the subgroup SU (2) always remain in such a subgroup. In fact consider the set of vector fields The integral curves for the t-dependent vector field (8.3) are solutions of the system Asa consequence of the standard methods developed for the theory of Lie systems [52], we join two copies of the above system in order to get the first-integrals {t, £0, 1, 2, 0212, v2}, reads
![Asa consequence of the standard methods developed for the theory of Lie systems [52], we join two copies of the above system in order to get the first-integrals](https://figures.academia-assets.com/78224965/figure_010.jpg)
