This is a joint work  with D. Dolgopyat. We consider a random walk (RW) in ergodic random environment (RE) on a strip in the 'environment viewed from the particle' setting. It is well known that in this setting the RW is a Markov chain on the set of environments. This approach to RWRE was introduced by S. Kozlov (1978, 1985) as well as Papanicolaou and Varadhan (1982) and the related fundamental question in this context is: Does this Markov chain have an invariant measure with a density with respect to the measure on the set of environments? It turns out that in the case of the walks on a strip in ergodic RE  it is possible to derive the necessary and sufficient conditions for the existence of the density in all regimes - transient and recurrent. We also answer some of the questions asked by Ya. Sinai in his paper "Simple random walks on tori" (J. Statist. Phys. 94 (1999), pp 695--708).