The XYZ Landau-Lifshitz (L-L) equation \[ S_ t=S\times S_{xx}+S\times JS,\quad | S| =1,\quad J=diag(J_ 1,J_ 2,J_ 3) \] is a compatibility condition for the set of two linear equations \[ \psi_ x=U(u)\psi,\quad \psi_ t=V(u)\psi \to U_ t-V_ x+[U,V]=0, \] where the spectral parameter U is a point on an elliptic curve T, and U,V are the (2\(\times 2)\) matrices. Precise formulae for the finite-gap (N-phase, algebro-geometric) solutions of the L-L equation are obtained. It is shown that the corresponding finite-gap \(\psi\)-function (Baker-Akhiezer function) is defined on the elliptic-hyperelliptic surface \(\Gamma\), i.e. a two-sheeted ramified covering of the torus \(T=\Gamma /\pi\) (the involution \(\pi\) transposes the sheets of \(\Gamma)\). The formulae for the \(\psi\)-function and the finite-gap solutions are expressed in terms of theta-functions. The remarkable fact is that the theta-functions are defined not by the Jacoby variety J(\(\Gamma)\), but the Prym variety \(\Pr ym_{\pi}(\Gamma)\) only. The connection with Clebsch's and Neumann's models is discussed. More convenient statement of results can be found in {\S} 7 of \textit{E. D. Belokolos}, the author, \textit{V. Z. Enolskij} and \textit{V. B. Matveev} [''Algebro-geometric principles of a superposition of finite-gap solutions of the integrable nonlinear equations'', Usp. Mat. Nauk 41, No.2, 3-42 (1986)]. By the method proposed the Euler equations on e(3) and SO(4) are integrated by the author [''Euler equations on e(3) and SO(4). Isomorphism of the integrable cases'', Funkts. Anal. Prilozh. 20, No.1, 64-66 (1986)].