Summary: We propose to analyse some classical overconstrained mechanisms by using explicitely the relations of singularity and not only the closure equation \(A_{1} \circ \cdots \circ A_{n} = e\). Using the fact that their configuration space manifold is such that the constraint Jacobian is permanently not of full rank, we apply our earlier developed analysis [the author, ibid. 78, No. 10, 687–694 (1998; Zbl 0910.70004)] of singularities to analyse overconstrained mechanisms. The holonomic constraints \(A_{1} \circ \cdots \circ A_{n} = e\) are linearized and formulated in an appropriate form on the Lie algebra. This gives a redundant set of linear equations which is solved. The used method is coordinate-free, and the solution of the Bennett and Bricard mechanisms is given as illustration of the method.