4 Integrability
In this section we derive the system (2.2) explicitly in terms of the functions \(A\), \(B\), \(C\) and \(D\) of Theorem 3.1. Moreover, we prove that it possesses a dispersionless Lax pair understood as a pair of commuting vector fields depending on a spectral parameter. Systems of this type, e.g., the dispersionless Kadomtsev-Petviashivili equation, often appear as dispersionless limits of integrable PDEs. Other examples include the Plebański heavenly equation or the Manakov-Santini system describing 3-dimensional Einstein-Weyl geometry. We refer to [15, 16] for general methods of integration of such systems. Let \(H\subset\mathrm{SL}(4,\mathbb{R})\) be the subgroup of matrices (3.1). Furthermore, let \(A_i\), \(B_i\), \(C_i\) and \(D_i\) denote \(\partial_iA\), \(\partial_iB\), \(\partial_iC\) and \(\partial_iD\), respectively,
An \(H\)-flat \(\mathrm{GL}(2)\)-structure defined by a coframing \(h\,\mathrm{d}x\), where \(h\) takes values in \(H\), is torsion-free if and only if \[\tag{4.1} \begin{aligned} &V_2(D)-V_3(B)-AV_2(B)-CV_2(A)+AV_3(A)+A^2V_2(A)=0\\ &2V_1(D)-V_2(C)-2AV_1(B)-V_3(A)+\\ &\qquad\qquad+AV_2(A)+2A^2V_1(A)-2CV_1(A)=0\\ &V_0(D)-2V_1(C)+3V_1(B)-AV_0(B)-2V_2(A)\\ &\qquad\qquad-AV_1(A)-CV_0(A)+A^2V_0(A)=0\\ &V_0(C)-2V_0(B)+V_1(A)+AV_0(A)=0, \end{aligned}\] and where the framing \((V_0,V_1,V_2,V_3)\) dual to \(h\,\mathrm{d}x\) is explicitly given by \[\begin{aligned} &V_0=\partial_0,\qquad V_1=\partial_1-A\partial_0,\qquad V_2=\partial_2-A\partial_1-(B-A^2)\partial_0,\\ &V_3=\partial_3-A\partial_2-(C-A^2)\partial_1-(D-(C+B)A+A^3)\partial_0. \end{aligned}\] The system (4.1) can be put in the Lax form \([L_0,L_1]=0\) with \[\begin{aligned} &L_0=\partial_3+(-C+2A\lambda-3\lambda^2)\partial_1\\ &\qquad+(-D+AC-2A^2\lambda+4A\lambda^2-2\lambda^3)\partial_0+\nu(\lambda)\partial_\lambda,\\ &L_1=\partial_2+(-A+2\lambda)\partial_1+(-B+A^2-2A\lambda+\lambda^2)\partial_0+ \mu(\lambda)\partial_\lambda \end{aligned}\] and \[\begin{aligned} \nu(\lambda)&=\left(\frac{1}{2}A^2A_1-ABA_0+AA_2-AB_1-\frac{1}{2}DA_0-\frac{1}{2}C_2\right.\\ &\qquad\qquad\left.+\frac{1}{2}AC_1+\frac{1}{2}BC_0-\frac{1}{2}CA_1+\frac{1}{2}ACA_0+\frac{1}{2}A_3\right)\\ &\phantom{=}+(3B_1-C_1-AA_1-AC_0+2BA_0-2A_2)\lambda\\ &\phantom{=}+(C_0-A_1)\lambda^2\\ \mu(\lambda)&=\left(\frac{1}{2}AA_1+\frac{1}{2}AC_0-BA_0+A_2-B_1\right)\\ &\phantom{=}+\left(\frac{1}{2}A_1-\frac{1}{2}C_0\right)\lambda,\end{aligned}\] for some auxiliary spectral coordinate \(\lambda\).
The spectral parameter \(\lambda\) can be treated as an affine parameter on the fibres of \(\mathcal{C}\). The theorem states that \(\mathcal{D}=\mathrm{span}\{L_0,L_1\}\) is an integrable rank-2 distribution on \(\mathcal{C}\). There is a 3-parameter family of integral manifolds of \(\mathcal{D}\). Projections of these submanifolds to \(M\) give a 3-parameter family of 2-dimensional submanifolds of \(M\) tangent to the field of cones \(\tilde{\mathcal{C}}\).
The space of integral manifolds of the aforementioned distribution \(\mathcal{D}=\mathrm{span}\{L_0,L_1\}\) is the twistor space \(T\) of a torsion-free \(\mathrm{GL}(2)\)-structure. In this context \(\mathcal{C}\) is the correspondence space and we have a double fibration picture \(M\longleftarrow \mathcal{C}\longrightarrow T\), where the fibres of the second projection are tangent to \(\mathcal{D}\). If the coefficients \(\mu\) and \(\nu\) in the Lax pair \((L_0,L_1)\) vanish, then there is additional natural projection, defined by the parameter \(\lambda\), from \(T\) to one-dimensional projective space. In other words, for any fixed \(\lambda\), the integral leaves of \(\mathcal{D}_\lambda=\mathrm{span}\{L_0(\lambda),L_1(\lambda)\}\) define a 2-dimensional foliation of \(M\). Among these structures there is a subclass for which the distribution \(\mathrm{span}\{L_0(\lambda),L_1(\lambda),\frac{d}{d\lambda}L_1(\lambda)\}\) is integrable and thus defines a 3-dimensional foliation. Such foliations are known as Veronese webs, c.f. [13]. From this point of view, the Veronese webs can be thought of as higher-dimensional counterparts of 3-dimensional hyper-CR Einstein-Weyl structures [5].
Veronese webs are described by a hierarchy of integrable systems introduced in [5], which generalize the dispersionless Hirota equation. It is worth seeing how the system (4.1) looks like in this case. For this we note that the \(H\)-flat form of 4-dimensional Veronese webs has been given in [14] and in this case we get (after permutation of indices) the following coefficients \[A=\frac{\partial _1f}{\partial _0f},\qquad B=C=\frac{\partial _2f}{\partial _0f},\qquad D=\frac{\partial _3f}{\partial _0f},\] where \(f=f(x_0,x_1,x_2,x_3)\) is a function. Then, in terms of \(f\), the system (4.1) takes the following simple form \[\begin{aligned} &f_2f_{00}-f_0f_{02}-f_1f_{01}+f_0f_{11}=0,\\ &f_3f_{00}-f_0f_{03}-f_1f_{02}+f_0f_{12}=0,\\ &f_3f_{01}-f_0f_{13}-f_2f_{02}+f_0f_{22}=0, \end{aligned}\] which coincides with the system derived in [14]. One can also set \(H_i=-\frac{f_{i+1}}{f_0}\) and pass to a system derived in [14]. An example of such a structure is given by the equation \(x^{(4)}=(x^{(3)})^{4/3}\) from [5]. In this case, using the formulae given in the proof of Theorem 3.1, one finds \(\alpha=x_0^{1/3}\), \(\beta=\gamma=x_0^{2/3}\) and \(\delta=x_0\). Thus \(A=-x_0^{1/3}\), \(B=C=D=0\) and \(f(x_0,x_1,x_2,x_3)=x_1-\frac{3}{2}x_0^{2/3}\).
A Cartan–Kähler analysis reveals that the first order system (4.1) – or equivalently (2.2) – is involutive and has solutions depending on four functions of three variables, confirming the count of Bryant [2]. Moreover, straightforward computations show that the characteristic variety of the system (4.1) linearised along any solution \((A,B,C,D)\) is the discriminant locus \(\mathcal{Q}\), i.e., the tangential variety of \(\mathcal{C}\).
Proof of Theorem 4.1. The system (4.1) can be directly obtained by expanding (2.2) explicitly in terms of the functions \(A,B,C,D\). Here we use a different method and apply [12] to the framing \((V_0,3V_1,3V_2,V_3)\). Namely, denoting \(\lambda=\frac{s}{t}\), we get that the curve \(\mathcal{C}\) in \(\mathbb{P}(TM)\) is the image of \(\lambda\mapsto \mathbb{R}V(\lambda)\in \mathbb{P}(TM)\), where \(V(\lambda)=\lambda^3 V_0+3\lambda^2 V_1+3\lambda V_2+V_3\) and the vector fields \(V_0\), \(V_1\), \(V_2\) and \(V_3\) are given by (3.2) with \[\alpha=-A,\quad\beta=-B+A^2,\quad\gamma=-C+A^2,\quad\delta=-D+(C+B)A-A^3.\] According to [12], a \(\mathrm{GL}(2)\)-structure is torsion-free if and only if \[\tag{4.2} \left[V(\lambda),\frac{d}{d\lambda}V(\lambda)\right]\in\mathrm{span}\left\{V(\lambda),\frac{d}{d\lambda}V(\lambda), \frac{d^2}{d\lambda^2}V(\lambda)\right\},\] for any \(\lambda\in\mathbb{R}\). This, due to [12] applied to the framing \[(V_0,3V_1,3V_2,V_3),\] is expressed as eight linear equations for structural functions \(c_{ij}^k\) defined by \([V_i,V_j]=\sum_kc_{ij}^kV_k\). However, in the present case, the vector fields \(V_i\) are special and four equations are void. Indeed, the nontrivial equations are as follows: \[\begin{aligned} c^0_{23}&=0, \qquad c^1_{23}-2c^0_{13}=0,\\ c^2_{23}-2c^1_{13}+c^0_{03}+3c^0_{12}&=0, \qquad c^3_{23}-2c^2_{13}+c^1_{03}+3c^1_{12}-2c^0_{02}=0\end{aligned}\] (the equations differ from equations in [12] because of the factor 3 next to \(V_1\) and \(V_2\) in the present paper). Substituting the structural functions, which can be easily computed, we get the system (4.1).
Now, we consider \[L_0=V(\lambda)-\left(\lambda-\frac{1}{3}A\right)\frac{d}{d\lambda}V(\lambda)\mod \partial_\lambda\] and \[L_1=\frac{1}{3}\frac{d}{d\lambda}V(\lambda)\mod \partial_\lambda.\] Due to (4.2), the commutator \([L_0,L_1]\) lies in the span of \(\{L_0,L_1,\frac{d^2}{d\lambda^2}V(\lambda)\}\mod\partial_\lambda\). Moreover, since \[L_0=\partial_3\mod\partial_1,\partial_0,\partial_\lambda \quad \text{and}\quad L_1=\partial_2\mod\partial_1,\partial_0,\partial_\lambda,\] we get \([L_0,L_1]=\varphi\frac{d^2}{d\lambda^2}V(\lambda)\mod\partial_\lambda\) for some \(\varphi\). One checks by direct computations that \(\mu(\lambda)\) and \(\nu(\lambda)\) are chosen such that \(\varphi=0\) and the coefficient of \([L_0,L_1]\) next to \(\partial_\lambda\) vanishes as well.