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In the present article, we develop the analysis of the following nonlinearelliptic system of equations $$ \bar\partial^\pi w = 0, \, d(w^*\lambda \circj) = 0 $$ first introduced by Hofer, associated to each given contact triad$(M,\lambda,J)$ on a contact manifold $(M,\xi)$. We directly work with thiselliptic system on the contact manifold without involving the symplectizationprocess. We establish the local a priori $C^k$ coercive pointwise estimates forall $k \geq 2$ in terms of $\|dw\|_{C^0}$ by doing tensorial calculations oncontact manifold itself using the contact triad connection introduced bypresent the authors. Equipping the punctured Riemann surface $(\dot \Sigma,j)$with a cylindrical K\"ahler metric and isothermal coordinates near everypuncture, we prove the asymptotic (subsequence) convergence to the `spiraling'instantons along the `rotating' Reeb orbit for any solution $w$, notnecessarily for $w^*\lambda \circ j$ being exact (i.e., allowing non-zero`charge' $Q \neq 0$), with bounded gradient $\|d w\|_{C^0} < C$ and finite$\pi$-harmonic energy. For nondegenerate contact forms, we employ the`three-interval method' to prove the exponential convergence to a closed Reeborbit when $Q = 0$. (The Morse-Bott case using this method is treated in asequel (arXiv:1311.6196).)

Analysis of Contact Cauchy-Riemann maps I: a priori $C^k$ estimates and asymptotic convergence