Critical points of a function subject to a constraint can be either detected by restricting the function to the constraint or by looking for critical points of the Lagrange multiplier functional. Although the critical points of the two functionals, namely the restriction and the Lagrange multiplier functional are in natural one-to-one correspondence this does not need to be true for their gradient flow lines. We consider a singular deformation of the metric and show by an adiabatic limit argument that close to the singularity we have a one-to one correspondence between gradient flow lines connecting critical points of Morse index difference one. We present a general overview of the adiabatic limit technique in the article [FW22b]. The proof of the correspondence is carried out in two parts. The current part I deals with linear methods leading to a singular version of the implicit function theorem. We also discuss possible infinite dimensional generalizations in Rabinowitz-Floer homology. In part II [FW22a] we apply non-linear methods and prove, in particular, a compactness result and uniform exponential decay independent of the deformation parameter.