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The present work has a double aim. On the one hand we call attention on therelationship existing between the Ashtekar formalism and othergauge-theoretical approaches to gravity, in particular the Poincar\'e GaugeTheory. On the other hand we study two kinds of solutions for the constraintsof General Relativity, consisting of two mutually independent parts, namely ageneral three-metric-dependent contribution to the extrinsic curvature $K_{ab}$in terms of the Cotton-York tensor, and besides it further metric independentcontributions, which we analyze in particular in the presence of isotropicthree-metrics.Comment: 10 Revtex pages, no figures. Modified versio

Towards a general solution of the Hamiltonian constraints of General Relativity