We investigate the integrability of generalized nonautonomous nonlinear Schrodinger (NLS) equations governing the dynamics of the single- and double-component Bose-Einstein condensates (BECs). The integrability conditions obtained indicate that the e
xistence of the nonautonomous soliton is due to the balance between the different competition features: the kinetic energy (dispersion) versus the harmonic external potential applied and the dispersion versus the nonlinearity. In the double-component case, it includes all possible different combinations between the dispersion and nonlinearity involving intra- and inter-interactions. This result shows that the nonautonomous soliton has the same physical origin as the canonical one, which clarifies the nature of the nonautonomous soliton. Finally, we also discuss the dynamics of two-component BEC by controlling the relevant experimental parameters.
