Bifurcated magnetohydrodynamic tokamak equilibrium states with axisymmetric or helical core structure are computed. When a peaked pressure profile is chosen, the helical core structures appear like the {em snakes} that are observed in the JET tokamak. They also have the allure of saturated ideal internal kinks. The existence of a magnetic island is not a requisite condition. Novel equilibrium states that can model the snake are obtained for a JET configuration when the $q$-profile has weak reversed magnetic shear with minimum $q$ values in the range of $0.94$ to $1.03$. At the lower end of this $q_{min}$ range, the equilibrium {em snake} structure lies radially well inside the domain for which $q_{min}leq 1$. Free boundary equilibria computed for the TCV tokamak develop helical cores when $\eta_N$ exceeds $0.3$ and have a significant axis excursion for $\eta_Ngeq 0.4$. At fixed $left < \eta ight>=1.6%$, the distortion of the magnetic axis is large in the range $0.95leq q_{min}leq 1.01$. The plasma-vacuum interface is not significantly altered by the internal helical deformations.