Perfect 2-colorings of Hamming graphs with eigenvalue $\lambda_2$

An eigenvalue of the perfect coloring is an eigenvalue of its parameter matrix. Perfect 2-colorings of Hamming graphs with eigenvalue $\lambda_1$ were classified earlier in [1].
In this paper we prove that every perfect 2-coloring of the graph $H(n, q)$ with eigenvalue $\lambda_2$ are reduced to perfect 2-colorings $H(3,q)$ by removing non-essential directions, except for colorings constructed from perfect 2-colorings of $H (2, q)$ by means of substitution swatches and colorings obtained from partitions of $H (4,2)$ into two cycles. A classification of perfect 2-colorings $H(n,q)$ with an eigenvalue $\lambda_2$ for $q = 2,3,4$ is found.
[1] A. D. Meyerowitz, Cycle-balance partitions for distance-regular graphs, Discrete Math 264:1-3 (2003), 149–165. https://doi.org/10.1016/S0012-365X(02)00557-5