Modified braid equations for SOq(3) and noncommutative spaces

General solutions of the RTT equation with a maximal number of free parameters in the specrtal decomposition of vector SOq(3) R matrices are implemented to construct modified braid equations (MBE). These matrices conserve the given, standard, group relations of the nine elements of T, but are not constrained to satisfy the standard braid equation (BE). Apart from q and a normalization factor our R contains two free parameters, instead of only one such parameter for deformed unitary algebras studied in a previous paper [Math. QA/0009178] where the nonzero right hand side of the MBE had a linear term proportional to (R(12)-R(23)). In the present case the rhs is, in general, nonlinear. Several particular solutions are given and the general structure is analyzed. Our formulation of the problem in terms of projectors yields also two new solutions of standard (nonmodified) braid equation which are further discussed. The noncommutative three-spaces obtained by implementing such generalized R matrices are studied. The role of coboundary R matrices (not satisfying the standard BE) is explored. The MBE and Baxterization are presented as complementary facets of the same basic construction, namely, the general solution of RTT equation. A new solution is presented in this context. As a simple but remarkable particular case a nontrivial solution of BE is obtained for q=1. This solution has no free parameter and is not obtainable by twisting the identity matrix. In the concluding remarks, among other points, generalization of our results to SOq(N) is discussed