A variant of the Monge-Cayley-Salmon theorem?

Setting aside the assumption that $\phi$ be a polynomial mapping for the moment (however, see below for a construction of a large family of polynomial solutions), if one makes the 'nondegeneracy' assumptions

1. $\mathrm{dim}\ \mathrm{span}\bigl( \phi_s(s,t), \phi_t(s,t)\bigr) =2 $,
2. $\mathrm{dim}\ \mathrm{span}\bigl( \phi_s(s,t), \phi_t(s,t), \phi_{ss}(s,t), \phi_{st}(s,t), \phi_{tt}(s,t) \bigr) = 3$ for all $(s,t)\in[0,1]^2$, and
3. the subspace $W(s,t) =\mathrm{span}\bigl( \phi_s(s,t), \phi_t(s,t), \phi_{ss}(s,t), \phi_{st}(s,t), \phi_{tt}(s,t) \bigr)\subset\mathbb{R}^4$ is not constant, in the sense that $W:[0,1]^2\to \mathrm{Gr}_3(\mathbb{R}^4)\simeq\mathbb{RP}^3$ has nonvanishing differential,

then one can show that the surface $\phi\bigl([0,1]^2\bigr)\subset\mathbb{R}^4$ is ruled (and does not lie in an affine $3$-space).

Such surfaces locally depend on three arbitrary functions of one variable in Cartan's sense. One way of describing them is this: Let $\Lambda$ be the space of (affine) lines in $\mathbb{R}^4$, a manifold of dimension $6$. Consider the ($9$-dimensional) bundle $\pi:F\to\Lambda$ whose fiber over $\lambda\in\Lambda$ is the flag variety of the $3$-dimensional vector space $\mathbb{R}^4/\lambda'$, where $\lambda'\subset\mathbb{R}^4$ is the linear subspace parallel to $\lambda$. Then there exists a smooth $4$-plane field $D\subset TF$ such that, if $\gamma\subset F$ is a generic curve tangent to $D$, then regarding $\gamma$ as a $1$-parameter family of affine lines via $\pi(\gamma)\subset \Lambda$, the union of these lines is a surface satisfying the above assumptions. (Here 'generic' means that the tangents to $\gamma$ do not lie in triple of hyperplanes in $D$.)

It is not hard to write down polynomial solutions: For example, if $f:\mathbb{R}\to\mathbb{R}^4$ is a polynomial curve that satisfies $f'(s)\wedge f''(s)\wedge f'''(s) \wedge f''''(s)\not = 0$, then the mapping $\phi:\mathbb{R}^2\to\mathbb{R}^4$ given by $$ \phi(s,t) = f(s) + f'(s)\, t $$ (which parametrizes the 'tangential development' of the curve $f$) satisfies these conditions when $t\not=0$. (By replacing $t$ by $t{+}1$, say, one could arrange that $\phi$ be an immersion on all of $[0,1]^2$.)

The following analysis is a more-or-less standard approach to verifying the above description using the so-called 'moving frame'. (I'm sure that the result itself is classical in some sense, though I don't know offhand where to look in the literature to find it.)

The three conditions listed above are actually independent of the choice of $st$-coordinates on the surface in $\mathbb{R}^4$ and so can be regarded as conditions on a surface $S\subset\mathbb{R}^4$ that, for local analysis purposes, can be taken to be smoothly embedded.

Let $B_0(S)$ denote the space of quintuples $(p;v_1,v_2,v_3,v_4)$ where $p$ lies in $S$ and the quadruple $(v_1,v_2,v_3,v_4)$ is a basis of $\mathbb{R}^4$ such that $(v_1,v_2)$ is a basis of $T_pS$ while $(v_1,v_2,v_3)$ is a basis of the $3$-dimensional subspace $W_pS\subset \mathbb{R}^4$. Then $B_0$ is a smooth submanifold of the product $\mathbb{R}^4\times\mathrm{GL}(4,\mathbb{R})$ that has dimension $2 + 4 + 3 + 4 = 13$.

It is useful to define $\mathbb{R}^4$-valued functions $x,e_i: \mathbb{R}^4\times\mathrm{GL}(4,\mathbb{R})\to\mathbb{R}^4$ such that $x(p;v_1,v_2,v_3,v_4) = p$ while $e_i(p;v_1,v_2,v_3,v_4) = v_i$. Then there exist unique (linearly independent) $1$-forms $\omega^i$ and $\theta^i_j$ on $\mathbb{R}^4\times\mathrm{GL}(4,\mathbb{R})$ such that (assuming the usual summation convention on repeated indices) the following structure equations hold: $$ \mathrm{d}x = e_i \omega^i \qquad\text{and}\qquad \mathrm{d}e_i = e_j\,\theta^j_i\,.\tag1 $$

$$ \mathrm{d}\omega^i = -\theta^i_j\wedge\omega^j \qquad\text{and}\qquad \mathrm{d}\theta^i_j = -\theta^i_k\wedge\theta^k_j\,.\tag2 $$

Now, pull back these functions and $1$-forms to $B_0(S)$ (but, as is customary, not notate the pullback). The definition of $B_0(S)$ and the assumptions on $S$ imply that $$ \mathrm{d}x\wedge e_1\wedge e_2 = \mathrm{d}e_1\wedge e_1\wedge e_2\wedge e_3 = \mathrm{d}e_2\wedge e_1\wedge e_2\wedge e_3 = 0 $$ while the two expressions $$ \bigl(\,\mathrm{d}e_1\wedge e_1\wedge e_2,\ \mathrm{d}e_2\wedge e_1\wedge e_2\bigr) \qquad\text{and}\qquad \mathrm{d}e_3\wedge e_1\wedge e_2\wedge e_3 $$ are nowhere vanishing. Using the above structure equations, these imply the relations $$ \omega^3 = \omega^4 = \theta^4_1 = \theta^4_2 = 0,\tag3 $$ while, because of the three assumptions, $\omega^1\wedge\omega^2$ is nonvanishing, the pair $(\theta^3_1,\theta^3_2)$ do not simultaneously vanish, and $\theta^4_3$ is nonvanishing.

Meanwhile, the structure equations yield $$ 0 = \mathrm{d}\omega^3 = -\theta^3_1\wedge\omega^1 -\theta^3_2\wedge\omega^2, $$ so there must exist functions $h_{ij}=h_{ji}$ for $1\le i,j,\le 2$, not all simultaneously vanishing, such that $\theta^3_i = h_{ij}\omega^j$. The quadratic form $h = h_{ij}\,\omega^i\omega^j$ is then nonvanishing and well-defined up to multiples on the surface $S$. Moreover, for $i = 1$ or $2$, $$ 0 = \mathrm{d}\theta^4_i = -\theta^4_k\wedge\theta^k_i = -\theta^4_3\wedge\theta^3_i\,. $$ Thus, since $\theta^4_3$ is nonvanishing, it follows that $\theta^3_1$ and $\theta^3_2$ are multiples of $\theta^4_3$. In particular, $\theta^3_1\wedge\theta^3_2$ vanishes identically, so $h_{11}h_{22}-{h_{12}}^2$ vanishes identically. Thus, the quadratic form $h$ has rank $1$.

Let $B_1(S)\subset B_2(S)$ denote the submanifold defined by $h_{11} = h_{12}=0$. It is a smooth submanifold of $B_0(S)$ of dimension $11$, and when all the forms and functions are pulled back to $B_1(S)$, we have $\theta^3_1 = 0$ while $\theta^3_2 = h_{22}\,\omega^2$. In particular, it now follows that $\theta^4_3$ is also a multiple of $\omega^2$, say $\theta^4_3 = f\,\omega^2$ for some $f$ (which is nonvanishing).

Moreover, $$ 0 = \mathrm{d}\theta^3_1 = -\theta^3_k\wedge\theta^k_1 = -\theta^3_2\wedge \theta^2_1 = -h_{22}\,\omega^2\wedge\theta^2_1, $$ so it follows (since $h_{22}$ is nonvanishing) that $\theta^2_1 = g\,\omega^2$ for some function $g$ on $B_1(S)$.

Now, $$ \mathrm{d}\omega^2 = -\theta^2_j\wedge\omega^j = -g\,\omega^2\wedge\omega^1 - \theta^2_2\wedge\omega^2 = -(\theta^2_2 - g\,\omega^1)\wedge\omega^2. $$ Thus, $\omega^2$ is an integrable $1$-form, and, because it is semi-basic for the submersion $x:B_1(S)\to S\subset\mathbb{R}^4$, it follows that $\omega^2$ is a multiple of the $x$-pullback of a (nonvanishing) $1$-form on $S$. Thus, $S$ is foliated by (connected) curves whose $x$-preimages in $B_1(S)$ are codimension $1$ integral submanifolds of $\omega^2$.

I claim that these curves in $S$ are, in fact, lines in $\mathbb{R}^4$. To see this, note that, when one pulls back to a leaf of $\omega^2=0$ in $B_1(S)$, one has $\theta^2_1 = \theta^3_1=\theta^4_1= 0$ as well, so one has $$ \mathrm{d}x = e_1\,\omega^1\qquad\text{and}\qquad \mathrm{d}e_1 = e_1\,\theta^1_1\,. $$ In particular, the direction of $e_1$ is fixed on this leaf, and this is the tangent direction of the mapping $x$ restricted to this leaf. Hence the $x$-image of this leaf is an open interval in a line in $\mathbb{R}^4$. Thus, the surface is ruled, as claimed.

There still remains the question of how one could 'generate' these surfaces, at least locally. I claim that, since these surfaces are, in fact, $1$-parameter families of lines, one should think of them as curves in the space of lines that satisfy some differential equations. Here is how one can think of this system of equations:

Let $\mathcal{I}$ denote the Pfaffian system of rank $5$ that is generated by the five linearly independent $1$-forms $$ \omega^3,\ \omega^4,\ \theta^4_1\,,\ \theta^4_2\,,\ \theta^3_1 $$ (these are the $1$-forms that vanish when pulled back to $B_1(S)$ when $S\subset\mathbb{R}^4$ is a surface satisfying our hypotheses). By the structure equations, $$ \left. \begin{aligned} \mathrm{d}\omega^4 &\equiv 0\\ \mathrm{d}\theta^4_1 &\equiv 0\\ \mathrm{d}\omega^3 &\equiv -\theta^3_2\wedge\omega^2\\ \mathrm{d}\theta^3_1 &\equiv -\theta^3_2\wedge\theta^2_1\\ \mathrm{d}\theta^4_2 &\equiv \phantom{-}\theta^3_2\wedge\theta^4_3\\ \end{aligned}\ \right\} \mathrm{modulo}\ \mathcal{I} $$ It follows that $\mathcal{J}$, the Cartan system of $\mathcal{I}$, has rank $9$ and is spanned by the nine $1$-forms $$ \omega^3,\ \omega^4,\ \theta^4_1\,,\ \theta^4_2\,,\ \theta^3_1\,,\ \omega^2,\ \theta^2_1\,,\ \theta^3_2\,,\ \theta^4_3 $$ In fact, $\mathcal{J}$ is easily seen to be the bundle of $1$-forms on $\mathbb{R}^4\times\mathrm{GL}(4,\mathbb{R})$ that is semibasic for the projection to the $9$-manifold $F$, where the projection sends $(p; v_i)$ to the element of $F$ described by $$ \bigl (\lambda(p,v_1),\ [v_1\wedge v_2], [v_1\wedge v_2\wedge v_3]\ \bigr) $$ and where $\lambda(p,v_1)$ is the line through $p$ in the direction $v_1$.

In particular $\mathcal{I}$ is the pullback of a well-defined Pfaffian system of rank $5$ on $F$, whose annihilator is the $4$-plane field $D\subset TF$. Moreover, it is easy to show that the three rank $6$ Pfaffian systems generated by adjoining any one of $\omega^2$, $\theta^3_2$, or $\theta^4_3$ to $\mathcal{I}$ are themselves pullbacks of rank $6$ Pfaffian systems on $F$ whose annihilators in $TF$ are each $3$-plane subbundles of $D$.

By its very construction, the projection of $B_1(S)$ into $F$ is a curve that is tangent to $D$ and not tangent to any of these three $3$-plane subbundles of $D$.

Conversely, if $\gamma\subset F$ is any integral curve of $D$ that is not tangent to any of the three $3$-plane subbundles of $D$, its preimage in $\mathbb{R}^4\times\mathrm{GL}(4,\mathbb{R})$ is a submanifold of the form $B_1(S)$ for a surface $S$ satisfying our conditions, in fact, the surface swept out by the union of the lines represented by $\lambda(\gamma)$, where $\lambda:F\to \Lambda$ is the obvious map to the lines.

It is a standard fact that the curves tangent to a 4-plane field in a smooth manifold are locally described (up to reparametrization) by prescribing $3$ functions of one variable.