I just read Chapter 17, on rational invariants, in Burnside's classic 1911 text Theory of Groups of Finite Order. It was a great read, and mostly it was straightforward to translate the ideas into modern language, but I was confused by one thing, which is what he means by "fixed systems". I will give the passage including his definition, as well as the context leading up to it.
Paragraph 263, which begins on p. 357, begins with a set $I_1,\dots,I_n$ of $n$ algebraically independent rational invariants of a degree-$n$ linear representation of a finite group $G$ of order $N$. (He says nothing about the ground field in this chapter but it's pretty clear from context that we can take it to be $\mathbb{C}$ or an algebraic closure of $\mathbb{Q}$. I'll go with $\mathbb{C}$ for the purposes of this post.) Burnside writes:
...consider the simultaneous equations $$I_r = a_r,\quad (r=1,2,\dots,n)$$ where the $a$'s are constants. If $$x_1=\alpha_1,x_2=\alpha_2,\dots,x_n=\alpha_n$$ is a solution of these equations, so also is $$x_1=\alpha_1^{(S)},x_2=\alpha_2^{(S)},\dots,x_n=\alpha_n^{(S)},$$ where $S$ is any substitution of the group. Two such solutions will be called "equivalent"; and the solutions that arise, when for $S$ is taken each substitution of the group in turn, will be called a system of equivalent solutions, or more shortly a "system."
So far, I'm totally on board. If $V$ is the representation space, so that $x_1,\dots,x_n$ are coordinate functions on $V$, then the choice of invariants $I_1,\dots,I_n$ specifies a rational map $\phi:V\rightarrow \mathbb{A}^n_\mathbb{C}$. Burnside fixes a point $(a_1,\dots,a_n)\in \mathbb{A}^n_\mathbb{C}$ and takes the fiber of $\phi$ over that point (if it is defined). Then $(\alpha_1,\dots,\alpha_n)\in V$ is a point of the fiber. The claim is that the fiber is a union of orbits of $G$ on $V$. "System" means orbit. Burnside continues:
In general $\alpha_1,\alpha_2,\dots,\alpha_n$ are algebraic functions of the $a$'s. A system of solutions for which this is the case will be called a variable system. The $n$ equations may however also admit systems of solutions which are independent of the $a$'s. Such systems will be called fixed systems. The number of distinct variable systems of equivalent solutions that the $n$ equations admit is necessarily finite; and, when different values are assigned to $a_1,a_2,\dots,a_n$, this number must have a greatest value $M$.
I'm puzzled by the "fixed systems". (The rest of it makes perfect sense to me: since the $n$ invariants are algebraically independent, the field extension $\mathbb{C}(V)/\mathbb{C}(I_1,\dots,I_n)$ is finite; $M$ is the degree.) I can't even parse the sentence, "The $n$ equations may however also admit systems of solutions which are independent of the $a$'s." The $n$ equations themselves depend on the $a$'s, so I am having trouble making sense of what is meant by saying that the solutions are independent of the $a$'s.
Can you articulate what Burnside is talking about in modern language?
