When it comes to a system of linear equations, - it is determined when it has $N$ variables and $N$ linearly independent equations.
If the equations are less, it is underdetermined; if more, it is overdetermined.
What about a system of linear PDEs?
Example: $f = f(x_1,x_2,...,x_N)$
The PDE system contains all first-order derivatives $\frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, ..., \frac{\partial f}{\partial x_N}$.
The PDE system is linear.
What conditions are to be met by the PDE system for it to have an unique solution?