For electronically degenerate states of linear tetra-atomic molecules, a new method is developed for the variational treatment of the Renner-Teller and spin-orbit couplings. The approach takes into account all rotational and vibrational degrees of freedom, the dominant couplings between the corresponding angular momenta as well as the couplings with the electronic and electron spin angular momenta. The complete rovibrational kinetic energy operator is expressed in Jacobi coordinates, where the rovibrational angular momenta JN have been replaced by J −Lez−S and the spin-orbit coupling has been described by the perturbative term ASO .Lez *S z. Attention has been paid on the electronic wave functions, which require additional phase for linear tetra-atomic molecules. Our implemented rovibrational basis functions and the integration of the different parts of the total Hamiltonian operator are described. This new variational approach is tested on the electronic ground state X 2Pi u of HCCH+ for which new six-dimensional potential energy surfaces have been computed using the internally contracted multireference configuration interaction method and the cc-pV5Z basis set. The calculated rovibronic energies and their comparisons with previous theoretical and experimental works are presented in the next paper.