  In this paper, we show that "Labyrinth walks", the conservative version of
"Labyrinth chaos" and member of the Thomas-R\"ossler class of systems, does not
admit an autonomous Hamiltonian as a constant function in time, and as a
consequence, does not admit a symplectic structure. However, it is
conservative, and thus admits a vector potential, being at the same time
chaotic. This exceptional set of properties makes "Labyrinth walks" an elegant
example of a chaotic, conservative, non-Hamiltonian system, with only unstable
stationary points in its phase space, arranged in a 3-dimensional grid. As a
consequence, "Labyrinth walks", even though is a deterministic system, it
exhibits motion reminiscent of fractional Brownian motion in stochastic
systems!
