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For a given generalized Nevanlinna function $Q\in N_{\kappa }\left( H\right)$, we study decompositions that satisfy: $Q=Q_{1}+Q_{2}$; $Q_{i}{\inN}_{\kappa_{i}}\left( H \right)$, and $\kappa_{1}+\kappa_{2}=\kappa $, $0\le\kappa_{i}$, which we call desirable decompositions. In this paper, somesufficient conditions for such decompositions of $Q$ are given.  One of the main results is a new operator representation of$\hat{Q}\left(z\right):=-{Q(z)}^{-1}$ if $Q\left( z\right):=\Gamma_{0}^{+}\left( A-z\right)^{-1}\Gamma_{0}$, where $A$ is abounded self-adjoint operator in a Pontryagin space. The new representation isused to get an interesting desirable decomposition of $\hat{Q}$ and to obtainsome information about singularities of $\hat{Q}$.

Desirable Decompositions of Generalized Nevanlinna Functions