Earlier we have found all 24 families of power-logarithmic expansions in p of solutions to the N.Kowalewski system of equations, describing motions of a rigid body with a fixed point in the case B ≠ C, x₀ ≠ 0, y₀=z₀=0. Among them, 10 families have p → 0 (tails) and 14 families have p → ∞ (heads). To find all finite expansions we check each pair a tail and a head: can it give a finite expansion or it cannot By this approach we find all finite solutions to the N.Kowalewski equations, in particular, all 7 known and 5 new. All new solutions are complex. We also prove the absente of other solution which is the finite sum of rational powers of p.

Mathematical problems and theory of numerical methods