Stability analysis and dimension of a dynamic control system
Differentiating $(3)$ with respect to time we get

$$u_{cc}\dot c = \dot \lambda \implies \frac {u_{cc}}{u_c} \dot c = \frac {\dot \lambda}{\lambda}$$

Inserting into $(6)$ we obtain

$$\dot M = -\frac {\eta_M}{\eta_{MM}}\frac {u_{cc}}{u_c} \dot c$$

So the fixed point of $M$ will happen under the same conditions that the fixed point of $c$ will. Also, the optimal time-evolution of $M$ is a scaled linear value of the optimal evolution of $c$. So it appears all aspects are captured if you "ignore" the $M$ variable.

A bit more formally, since your two decision variables are linearly dependent it means that the $3 \times 3$ Jacobian of the system will be singular at the fixed point. You will get a double eigenvalue and a single one, and you should be able to show what stability properties hold (or what is required for the desired stability notion to hold), in such a situation. Due to the existence of the double root, this requires a bit of a different mathematical approach.