This article contains a proof of an important theorem in soliton mathematics. The theorem, stated roughly in [4], contains necessary conditions for the existence of a vector function \[ ψ ( t , p ) = ( ψ 1 ( t , p ) , … , ψ l ( t , p ) ) , t ∈ C g , p ∈ R , \psi (t,p) = ({\psi _1}(t,p), \ldots ,{\psi _l}(t,p)),\quad t \in {{\mathbf {C}}^g},\quad p \in R, \] with prescribed poles and l l essential singularities an a compact Riemann surface R R of genus g g . ψ \psi is called a Baker function in honor of the 1928 article [1] of H. F. Baker. This report clarifies Krichever’s description of ψ \psi for l > 1 l > 1 essential singularities. The divisors δ α {\delta _\alpha } in (1) below are the key to the l > 1 l > 1 construction. Krichever’s ( l > 1 ) (l > 1) construction is not easy to deal with in practical problems. E. Previato [5] noted this and applied our characterization of the δ α {\delta _\alpha } to construct the finite gap solutions to the nonlinear Schroedinger equation.