On matrix integrability

I will present the results of our joint work with A. Guterman, S. Danielyan and F. Pakovich, https://arxiv.org/pdf/2303.13239.pdf
A matrix integral is a matrix larger by one, in which the original matrix is “inscribed” in the upper left corner, and whose characteristic polynomial is proportional to the integral of the characteristic polynomial of the original matrix. Not every matrix possess an integral, and the question of the existence of integrable and non-integrable matrices with a prescribed Jordan structure arises. It turned out that both of these questions (in different ways) are reduced to the question of the existence of two-colored trees with certain properties of passports. Namely, it appears that Shabat polynomials are responsible for the existence of integrable matrices, while conservative polynomials are responsible for the existence of non-integrable matrices.