Calculation of first- and second-order symmetries in Jaynes-Cummings model

We provide three files created in the symbolic computation software Maple (licensed to Universidad de Guadalajara), which were used to calculate the higher symmetries of the system of differential equations associated with the Jaynes-Cummings model (see ArXiV https://arxiv.org/abs/2606.15538). The file Jaynes_CL_2025_2026_NEW_LINEAR_v6_c1_-1.mw contains a semi-automated process for finding the generating functions for first- and second-order symmetries (see Steps to reproduce for details). File verification_Heun_polynomial_final.mw verifies the fulfillment of the sufficient condition for the confluent Heun function to be a polynomial, given the parameters specified in the article. File Luis_graficas_Heun_2025_2026.mw handles the graphics of Heun polynomials and radial components for both invariant solutions; it is also used to generate 3D plots of the characteristic functions at the initial time.