The analytical solutions of a nonlinear fin problem with variable thermal conductivity and heat transfer coefficients are investigated by considering theory of Lie groups and its relations with lambda-symmetries and Prelle-Singer procedure. Additionally, the classification problem with respect to different choices of thermal conductivity and heat transfer coefficient functions is carried out. In addition, Lagrangian and Hamiltonian forms related to the problem are investigated. Furthermore, the exact analytical solutions of boundary-value problems for the nonlinear fin equation are obtained and represented graphically. (C) 2020 The Authors. Published by Atlantis Press B.V.