Suppose we have a function $f(x_1,\ldots,x_n)$ that we wish to maximize under the set of $n-1$ constrictions $g_i(x_1,\ldots,x_n) = c_i$ for $i \in \{1,\ldots,n-1\}$.
We write the Lagrangian $L(x,\lambda) = f(x) + \sum_i^{n-1} \lambda_i (g_i(x) - c_i)$, and now we equate all partial derivatives with respect to $x_i, \lambda_i$ to $0$.
Let us look only on the equations $\frac{\partial L}{\partial x_i} = 0$. We have $n$ of those, and they feature $(n-1)$ $\lambda 's$. So we could, in theory, at least locally, look at $(n-1)$ first equations, and from them express $\lambda_i = \lambda_i(x_1,\ldots,x_n)$. We could substitute these into the $n$'th equation, and get: $$ \frac{\partial L}{\partial x_n} = \hat G(x_1,\ldots,x_n,\lambda_1(x_1,\ldots,x_n),\ldots,\lambda_n(x_1,\ldots,x_n)) = G(x_1,\ldots,x_n) = 0 $$
I.e. we get of an equation of a hypersurface (which is independent of $c_i$). My question is, is there a meaning to this manifold? I think I can see its intuitive explanation for $x = (x_1,x_2)$ and one constriction, but I don't really see it as clearly for a general $n$.
EDIT: Actually, the same could be repeated for $n-m$ constraints, and the result will similarly by an $(n-m)$-dimensional manyfold.
For what's it worth, I'm answering my own question. Here's what I think, after some time; The answer is more simple that what I'd have expected, but was actually useful to me.
The surface is a surface of points that are critical points for some values of $c_i$.
I.e. if $G(x_1,\ldots,x_n) = 0$, then $ \exists~ c_1, \ldots,c_{n-1}$ such that $g_i(x) = c_i$, and $x$ is a critical point of $f$ under these constraints.
(The reason is that given such an $x$, we can find the $c_i$ by solving for them, and then this $x$ will be a solution of the full system of the lagrangian, hence a critical points under the constrains.)
In case of $n=2$ and a single constraint, this can be readily visualized: We have a continuous family of curves that are the level sets of $g$. If on each curve we find the local maximum of $f$, and mark it with a red dot, then the red dots will themselves form a curve, whose equation will be the curve in the question.