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We present the new, general, explicit form of the equations of motion for constrained mechanical systems applicable to systems with singular mass matrices. The systems may have holonomic and/or non-holonomic constraints, which may or may not satisfy D'Alembert's principle at each instant of time. The equation provides new insights into the behaviour of constrained motion and opens up new ways of modelling complex multi-body systems. Examples are provided and applications of the equation to such systems are illustrated.