In Recursive predicates and quantifiers (Trans. Amer. Math. Soc. 53 (1943), 41-73) Kleene gives a description of general recursive functions acording to Herbrand and Gödel, 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 *no* restriction on how the function symbols appear, other than their arities being respected. There are basic substitution rules for manipulating expressions, see the paper.

It's not at all clear how to deal with a system of *arbitrary* equations. It is not so surprising that the definition of general recursive functions was considered unsatisfactory at the time.

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.