The KPZ fixed point is a Markov process on the space of upper semi-continuous functions. It is the conjectured universal scaling limit of the height function evolution for models in the KPZ universality class and has been shown to be such for many solvable (and even some unsolvable) models. The directed landscape provides a coupling for the growth of the KPZ fixed point starting from all initial conditions. Under this coupling, starting from an initial condition, the forward evolution of the KPZ fixed point can be described by a variational problem involving the directed landscape. It is an interesting question to characterize all eternal solutions of the KPZ fixed point (i.e.\ functions defined for both forward and backward times and satisfying the variational formula at all times), and in this talk we focus on this question.

One of our results give a full characterization of eternal solutions whose asymptotic slope is the same in the positive and negative directions. A key feature of the KPZ fixed point evolution is that the asymptotic slope is conserved. For a fixed realization of the directed landscape, it is known that, with probability one, there exists a random set of exceptional slopes. For slopes outside this set, the eternal solution with the prescribed slope is unique, whereas for exceptional slopes there are at least two such solutions. We give a full description of the eternal solutions with these exceptional slopes. In particular, we show that each exceptional slope admits uncountably many eternal solutions. The talk is based on joint work with Ofer Busani and Evan Sorensen.