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Spectral Meromorphic Operators and Nonlinear Systems

We study here class of 1D spectral-meromorphic (s-meromorphic) OD operators $L=\partial_x^n+\sum_{n-2\geq i\geq 0}a_{n-2-i}\partial_x^i$ with meromorphic coefficients $a_j$ near $x\in R$ such that all eigenfunctions $Lψ=αψ$ are $x$--meromorphic near $x\in R$ for all $α$. Symmetric $s$-meromorphic operators are self-adjoint with respect to indefinite inner product well-defined for some special spaces of singular functions. In particular, all algebraic operators $L$--i.e. operators entering Burchnall-Chaundy-Krichever (BChK) rank one commutative rings -- are s-meromorphic. For KdV system corresponding algebraic operator $L=-\partial_x^2+u(x,t)$ is called singular finite gap, singular soliton or algebrogeometric Schrodinger operator. This special case was already studied by the present authors in the recent works.