In [Recursive predicates and quantifiers](https://doi.org/10.2307/1990131) (Trans. Amer. Math. Soc. 53 (1943), 41-73) Kleene gives a description of general recursive functions, as understood before the paper "standardized" the subject.

According to the definition given therein (I am skipping over some details involving *given*, *auxiliary*, and *principal* function symbols), a *sytem* of equations $\mathcal{S}$ in function symbols $f_1, \ldots, f_n$, each with a given arity, is a list of equations of the form $L = R$, where $L$ and $R$ are expressions obtained from $0$, successor, variables, and the function symbols $f_i$. There is on restriction on how the function symbols appear, other than their arities being respected.

We have basic rules of deduction for reasoning about systems of equations, see the paper. They are just obvious substitution rules. It's easy to imagine that it's not so clear how to deal with a system of *arbitrary* equations, which is presumably why the definition was not considered satisfactory.

Kleene gives an example showing hot to express the minimization operator $\mu y . (r(x_1, \ldots, x_n, y) = 0)$ in this way (I write $\vec{x}$ for $x_1, \ldots, x_n$ and $n^{+}$ for successor of $n$):
\begin{align*}
\sigma(0, \vec{x}, y) &= y\\
\sigma(z^{+}, \vec{x}, y) &= \sigma(r(\vec{x}, y^{+}), \vec{x}, y^{+}) \\
\phi(\vec{x}) &= \sigma(\rho(\vec{x},0), \vec{x}, 0)
\end{align*}
This is a system of equations in $\sigma$ and $\phi$, for a given $r$. We then set $\mu y. (r(\vec{x}, y) = 0) \mathrel{{:}{=}} \phi(\vec{x})$.

Kleene goes on to prove that one can reduce every system $\mathcal{S}$ to one involving primitive recursive functions (which he defines in the usual way) and the minimization operator $\mu$. Apparently after this paper everyone forgot about the original unruly definition in terms of systems of arbitrary equations.