Degenerated differential equations with variable degeneration operator

The theory of Jordan chains for multiparameter operator-functions $A(\lambda):E_{1}\rightarrow E_{2}$, $\lambda\in\Lambda$, $\text{dim}\Lambda=k$, $\text{dim}\, E_{1}=\text{dim}\, E_{2}=n$ is developed. Here $A_{0}=A(0)$ is degenerated operator, $\mathit{dim}\mathit{Ker}A_{0}=1,$ $\mathit{Ker}A_{0}=\{\varphi\}$, $\mathit{Ker}\text{A}_{0}^{*}=\{\psi\}$ and the operator-function $A(\lambda)$ is supposed to be linear on $\lambda$.The aims of the article are the applications to degenerate differential equations of the form $[A_{0}+R(\cdot,x)]x'=Bx$.