The theory of the seventh-order aberrations of gradient-index media with axial symmetry is presented. This theory is based on Hamilton's equations. Formulas for seventh-order aberrations are derived. On the basis of the derived formulas the explicit algebraic expressions for all seventh-order aberration coefficients for gradient-index media with cylindrical symmetry are evaluated. The correctness of the obtained coefficients is verified for secant hyperbolic exact and helical exact gradient-index distributions. It is also shown that, for meridional rays, some of the aberration coefficients vanish for other index distributions besides the secant hyperbolic exact gradient-index distribution.