Biharmonic Helices in Three-Dimensional Lorentzian Egorov and $\varepsilon$-Spaces

We consider three-dimensional Lorentzian Egorov and $\varepsilon$-spaces and show that the existence of proper biharmonic helices in these manifolds is highly constrained and closely tied to the causal character of the Frenet frame. In particular, we prove that no proper biharmonic helices exist in Egorov spaces unless the space is flat and the normal vector is spacelike. Furthermore, we establish that no proper biharmonic helices exist in three-dimensional $\varepsilon$-spaces.