"Roughly", Nash embedding theorem is about the existence of $m$ (to be chosen) functions $x_i$ (of $n$ $u_j$) with some required order of differentiability and solutions of $$\sum_{i=1}^{m} \frac{\partial x_i}{\partial u_j} \frac{\partial x_i}{\partial u_k} = g_{jk} \qquad \forall j,k \in \{1,2,\ldots,n\}:\ \ j\leq k$$ where $g_{jk}$ are the components of a Riemannian metric and the $u_j$'s are coordinates of a point in the manifold.
The local real-analytic version of the Nash Embedding Theorem as established by Janet, Cartan, or Burstin gives, for $m=n(n+1)/2$, an analytic solution.
I am interested in a variation on this problem where one of the quadratic equations above is replaced by a linear one. Precisely
Question: Does there exist $m$ (to be chosen) $C^1$ functions $x_i$ solutions of $$ \sum_{i=1}^{m} \frac{\partial x_i}{\partial u_j} \frac{\partial x_i}{\partial u_k} = g_{jk} \qquad \forall j \in \{1,2,\ldots,n\} , \quad \forall k \in \{2,3,\ldots,n\} :\ \ j\leq k $$ and $$\sum_{i=1}^{m} a_i x_i = u_1$$ where $a_i$ are constants free to choose.
Note that the equation for $g_{11}$ is not in the list of quadratic equations. It is replaced by the (last) linear equation
Motivation: We want the hyperplanes $u_1$=constant to be totally geodesic in $\mathbb{R}^n$ equipped with some given metric $g$ and coordinates $(u_i)$. If this does not hold, I want this to be true in $\mathbb{R}^m$ ($m>n$) equipped with a metric which is an image of $g$ by an embedding $x$. The quadratic equations above are obtained by imposing further that this image metric is Euclidean (except for $g_{11}$). The linear equation just says that the image by $x$ of $u_1$=constant (in $\mathbb{R}^n$) is an hyperplane in $\mathbb{R}^m$.