Suppose in each equation, not counting covariance restrictions, we need one more restriction to meet the order condition. If we now add to each equation a restriction that its structural residual is uncorrelated with the residual of some other equation, is the parameter of the new model identifiable globally? That is the question. In general the answer is no. The parameter could remain either not identifiable or is locally identifiable, possibly globally under additional inequality restrictions. In this paper we find families of models for which the answer to the question is yes without the help of inequalities. The families share common characteristics. First, the sufficient condition for local identifiability must hold. Secondly, the string of zero correlations between residuals contains a closed cycle of length at least four. Thirdly, with the variables, equations and residuals all numbered as they are in the cycle, the odd numbered variables must satisfy a kinship relationship and lastly, the structural residuals can not all be uncorrelated. There are also differences in families, but these come from the difference in the required kinship relationship. When there are four or more equations containing external variables, the variety of models with uniquely identifiable parameter under a string of uncorrelated residuals is considerable. In particular, when correlated inverse demand shocks are uncorrelated with correlated supply shocks, our results show that many flexible inverse demand and supply equations reproducing exactly the observed price and quantity moments are members of the above families.