  This paper is the first of a three part series in which we explore geometric
and analytic properties of the Kohn Laplacian and its inverse on general
quadric submanifolds of $\mathbb{C}^n\times\mathbb{C}^m$. In this paper, we
present a streamlined calculation for a general integral formula for the
complex Green operator $N$ and the projection onto the nullspace of $\Box_b$.
The main application of our formulas is the critical case of codimension two
quadrics in $\mathbb{C}^4$ where we discuss the known solvability and
hypoellipticity criteria of Peloso and Ricci \cite{PeRi03}. We also provide
examples to show that our formulas yield explicit calculations in some
well-known cases: the Heisenberg group and a Cartesian product of Heisenberg
groups.
