  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
$L\psi=\alpha\psi$ are $x$--meromorphic near $x\in R$ for all $\alpha$.
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.
