The connection of the nonlinear equations theory with the spectral theory of operators and algebraic geometry of Riemannian surfaces and abelian manifolds is illustrated in this article on the example of \(n\times n\)- operators \(L(\lambda)=i\partial_ x+\lambda A-U(x)\) and the nonlinear equations associating with them (here \(A=diag(a_ 1,...,a_ n)\) is a real diagonal matrix with in pairs nonequal diagonal elements \(a_ 1,...,a_ n\); \(\lambda\) is a spectral parameter; U(x) is smooth periodic or quasiperiodic potential satisfying the J-autoconjugate condition \(U^*=JUJ\), \(J=diag(\pm 1,...,\pm 1))\). In particular, the conditions for the solution of the inverse spectral problem for finite zone operators L(\(\lambda)\) are discussed, the problem of classification of these operators is reduced to the problem of real algebraic geometry with topological classification of the triples of the spectral data (\(\Gamma\),\(\lambda\),\(\tau)\).