Two classes of multi-phase algebro-geometric solutions of Mikhalëv equation are constructed. The first class of solutions is associated with the Korteweg-de-Vries (KdV) equation. The second one is related to the solutions of the Kaup-Boussinesq (KB) equation. We have established interrelations among the multi-soliton, trigonometric, rational, elliptic and other known solutions of the KdV and KB equations and the solutions of Mikhalëv equations. We show that the number of linearly independent finite-gap solutions of Mikhalëv system is equal to the number of phases of these solutions. For each class of solutions we have constructed examples of explicit solutions of Mikhalëv equation. In the previous works cited below the solutions of the Mikhalëv system were described implicitly, being reduced to the solutions of appropriate Jacobi inversion problems. Here, to solve the Mikhalëv system explicitly, we used the formalism of Baker-Akhiezer functions.