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The principal objective in this paper is a new inverse approach to generalDirac-type systems modeled after B. Simon's 1999 inverse approach to half-lineSchr\"odinger operators. In particular, we derive the so-called A-equationassociated to Dirac-type systems and, given a fundamental positivity condition,we prove that this integro-differential equation for A is uniquely solvable. Weshow how to recover the matrix-valued potential coefficient from A. Thisapproach is also applicable to matrix-valued Schr\"odinger operators on ahalf-line.

The inverse approach to Dirac-type systems based on the $A$-function concept