Magnetohydrodynamic (MHD) equilibrium states with a three-dimensional helical core that display the characteristics of a saturated ideal internal kink mode are computed to model snake structures that have been observed in the JET tokamak [A.~Weller et al. 1987 Phys.~Rev.~Lett.~ extbf{59} 2303]. The equilibrium states are calculated with a peaked pressure profile and a weak to moderate reversed core magnetic shear with a minimum safety factor $q_{min}$ near unity in the neighbourhood of the mid-radius of the plasma. Snake equilibrium states are computed in the range $0.94<q_{min}<1.03$. This range aligns with linearly unstable ideal MHD internal kink solutions of the purely axisymmetric branch of the equilibrium states. The energy difference between the bifurcated axisymmetric and helical snake equilibrium solutions is minimal. One very important novelty is that the helical structures are computed with an equilibrium code developed for 3D stellarator applications in a tokamak context and cannot be obtained with standard Grad-Shafranov equation solvers.