Integrable systems with BMS$_{3}$ Poisson structure and the dynamics of locally flat spacetimes

We construct a hierarchy of integrable systems whose Poisson structure corresponds to the BMS$_{3}$ algebra, and then discuss its description in terms of the Riemannian geometry of locally flat spacetimes in three dimensions. The analysis is performed in terms of two-dimensional gauge fields for $isl(2,R)$. Although the algebra is not semisimple, the formulation can be carried out \`a la Drinfeld-Sokolov because it admits a nondegenerate invariant bilinear metric. The hierarchy turns out to be bi-Hamiltonian, labeled by a nonnegative integer $k$, and defined through a suitable generalization of the Gelfand-Dikii polynomials. The symmetries of the hierarchy are explicitly found. For $k\geq 1$, the corresponding conserved charges span an infinite-dimensional Abelian algebra without central extensions, and they are in involution; while in the case of $k=0$, they generate the BMS$_{3}$ algebra. In the special case of $k=1$, by virtue of a suitable field redefinition and time scaling, the field equations are shown to be equivalent to a specific type of the Hirota-Satsuma coupled KdV systems. For $k\geq 1$, the hierarchy also includes the so-called perturbed KdV equations as a particular case. A wide class of analytic solutions is also explicitly constructed for a generic value of $k$. Remarkably, the dynamics can be fully geometrized so as to describe the evolution of spacelike surfaces embedded in locally flat spacetimes. Indeed, General Relativity in 3D can be endowed with a suitable set of boundary conditions, so that the Einstein equations precisely reduce to the ones of the hierarchy aforementioned. The symmetries of the integrable systems then arise as diffeomorphisms that preserve the asymptotic form of the spacetime metric, and therefore, they become Noetherian. The infinite set of conserved charges is recovered from the corresponding surface integrals in the canonical approach.

One of the main purposes of our work is exploring whether the BMS 3 algebra could be further linked with some sort of integrable systems. There are some hints that can be borrowed from CFTs in two dimensions that suggest to look towards this direction. Indeed, it is known that CFTs in 2D admit an infinite set of conserved charges that commute between themselves, which can be constructed out from suitable nonlinear combinations of the generators of the Virasoro algebra and their derivatives (see e.g. [67]). Remarkably, these composite operators turn out to be precisely the conserved charges of the KdV equation, which also correspond to the Hamiltonians of the KdV hierarchy. Therefore, since the BMS 3 algebra can be seen as a limiting case of the conformal algebra in 2D, it is natural to wonder about the possibility of performing a similar construction in that limit. Specifically, one would like to know about the existence of an infinite number of commuting conserved charges that could be suitably recovered from the BMS 3 generators, as well as its possible relation with some integrable system, or even to an entire hierarchy of them. Here we show that this is certainly the case.
Furthermore, and noteworthy, the dynamics of this class of integrable systems can be equivalently understood in terms of Riemannian geometry. Indeed, following similar strategy as the one in [68], here we show that General Relativity without cosmological constant in 3D can be endowed with a suitable set of boundary conditions, so that in the reduced phase space, the Einstein equations precisely agree with the ones of the hierarchy aforementioned. In other words, the dynamics of our class of integrable system can be fully geometrized, since it can be seen to emerge from the evolution of spacelike surfaces embedded in locally flat spacetimes. As a remarkable consequence, in the geometric picture, the symmetries of the integrable systems correspond to diffeomorphisms that maintain the asymptotic form of the spacetime metric, so that they manifestly become Noetherian. Hence, the infinite set of conserved charges can be readily obtained from the surface integrals associated to the asymptotic symmetries in the canonical approach.
In the next section we explicitly construct dynamical (Hamiltonian) systems whose Poisson structure corresponds to the BMS 3 algebra. In order to analyze their symmetries and conserved charges, in section 3 we show how the Drinfeld-Sokolov formulation can be adapted to our case, through the use of suitable flat connections for isl(2, R). Section 4 is devoted to a thorough construction and the analysis of an entire bi-Hamiltonian hierarchy of integrable systems with BMS 3 Poisson structure, labeled by a nonnegative integer k. We start with a very simple dynamical system (k = 0) from which the bi-Hamiltonian structure can be naturally unveiled. The case of k = 1 is described in section 4.2, where the Abelian infinite-dimensional symmetries and conserved charges are explicitly identified in terms of a suitable generalization of the Gelfand-Dikii polynomials (see also appendices A and B). The equivalence between our field equations and the ones of the Hirota-Satsuma coupled KdV system of type ix, is shown to hold by virtue of a suitable field redefinition and time scaling in section 4.2.4. The hierarchy of integrable systems with BMS 3 Poisson structure is then explicitly discussed in section 4.3, where it is shown that the so-called "perturbed KdV equations" are included as a particular case. Section 4.4 is devoted to the construction of a wide interesting class of analytic solutions for a generic value of the label of the hierarchy k. In section 5, we show how the dynamics of the hierarchy of integrable systems can be fully geometrized in terms of locally flat three-dimensional spacetimes. The deep link with General Relativity in 3D is explicitly addressed. We conclude with some remarks about possible extensions of our results in section 6. Appendices A, B and C are devoted to some technical remarks.

Dynamical systems with BMS 3 Poisson structure
In order to construct dynamical systems whose Poisson structure is described by the BMS 3 algebra, let us consider two independent dynamical fields, J = J (t, φ) and P = P(t, φ), being defined on a cylinder whose coordinates range as 0 ≤ φ < 2π, and −∞ < t < ∞. The Poisson structure we look for can then be defined in terms of the following operator where D (J ) and D (P) stand for Schwarzian derivatives, given by with arbitrary constants c J and c P . The operator D (2) in eq. Therefore, the brackets of the dynamical fields read so that expanding in Fourier modes as X = 1 2π n X n e −inφ , the algebra of the Poisson brackets in (2.4) precisely reduces to the BMS 3 algebra in eq. (1.1).
The field equations for the class of dynamical systems we were searching for can then be readily defined as follows J Note that unwrapping the angular coordinate to range as −∞ < φ < ∞, allows to extend this class of dynamical systems to R 2 , provided that the fall-off of the dynamical fields J , P is sufficiently fast in order to get rid of boundary terms. Hereafter, for the sake of simplicity, we will assume that the dynamical systems are defined on a cylinder, with a single exception for an interesting particular solution that is described in section 4.4.2.

Zero-curvature formulation
In order to study the properties of the dynamical systems with BMS 3 Poisson structure that evolve according to eq. (2.5), including their symmetries and the corresponding conserved charges, it turns out to be useful to reformulate them in terms of a flat connection for a suitable Lie algebra (see e.g. [69,70]). In the standard approach of Drinfeld and Sokolov [71], the Lie algebra is assumed to be semisimple. Here we slightly extend this approach in a sense explained right below. For our purposes, the relevant Lie algebra to be considered corresponds to isl(2, R), which is isomorphic to the Poincaré algebra in 3D. Their commutation relations can then be written as where J a stand for the generators of sl(2, R) so(2, 1). It is worth emphasizing that the algebra (3.1) is not semisimple; but nonetheless, it admits a non degenerate invariant bilinear metric whose nonvanishing components read where c J and c P are arbitrary constants. 1 Hence, by virtue of (3.2), the analysis of the class of dynamical systems defined in section 2 can still be performed à la Drinfeld-Sokolov, provided that the field equations are able to be reproduced in terms of a flat connection for isl(2, R).
We then propose that the spacelike component of the isl(2, R)-valued gauge field a = a µ dx µ is given by with J = J (t, φ) and P = P(t, φ), while the timelike component reads Here µ J and µ P can be assumed to be given by some arbitrary functions of J , P, and their derivatives along φ (being denoted by a prime here and afterwards). Therefore, requiring the field strength for the gauge field defined in (3.3) and (3.4) to vanish, i.e., implies that the field equations for the dynamical system with BMS 3 Poisson structure in (2.5) hold. It then goes without saying that a φ and a t correspond to a Lax pair.

Symmetries and conserved charges
One of the advantages of formulating the field equations in terms of a flat connection, is that the symmetries of the dynamical system in (2.5) turn out to correspond to gauge transformations, δ λ a = dλ + [a, λ], that preserve the form of the gauge field defined through (3.3) and (3.4). Hence, requiring the form of the spacelike component of the connection a φ in (3.3) to be preserved under gauge transformations, restricts the Lie-algebra-valued parameter to be of the form where Λ is precisely given by eq. (3.5), but now depends on two arbitrary functions ε J = ε J (t, φ) and ε P = ε P (t, φ), while the dynamical fields have to transform according to Analogously, preserving the timelike component of the gauge field a t in (3.4), implies that the transformation law of the functions µ J and µ P is given by However, µ J and µ P generically depend on the dynamical fields and their spatial derivatives, which means that eqs. (3.9), (3.10) actually become a consistency condition to be fulfilled by the functions ε J and ε P that parametrize the symmetries of the dynamical system. In the case of Hamiltonian systems, µ J and µ P are determined by the corresponding functional variations of the Hamiltonian as in (2.6), and consequently, the consistency condition for the functions ε J and ε P , that arises from (3.9), (3.10), can be compactly written as In sum, the functions that parametrize the symmetries of the Hamiltonian system with BMS 3 Poisson structure must fulfill the consistency condition in (3.11), which for an arbitrary choice of Hamiltonian, implies that ε J and ε P generically acquire an explicit dependence on the dynamical fields and their spatial derivatives.
The variation of the canonical generators associated to the symmetries spanned by ε J , ε P can then be readily found by virtue of eqs. (3.2), (3.3) and (3.7), which reduces to the following simple expression (3.12) As a cross-check, it is simple to verify that the variation of the canonical generators is conserved (δQ = 0) provided that the consistency condition for the symmetry parameters in (3.11) is satisfied. Nevertheless, it must be highlighted that finding the explicit form of the conserved charges Q is not so simple, because it amounts to know the general solution of the consistency condition for the parameters in (3.11). Indeed, although the consistency condition is linear in the parameters ε J , ε P , the generic solution manifestly depends on the dynamical fields and their spatial derivatives that fulfill a nonlinear field equation. Thus, for a generic choice of the Hamiltonian, solving eq. (3.11) is actually an extremely difficult task.
However, for a generic Hamiltonian that is independent of the coordinates, the conserved charges associated to translations along space and time can be directly constructed. In fact, for a flat connection, diffeomorphisms spanned by ξ = ξ µ ∂ µ are equivalent to gauge transformations generated by λ = −ξ µ a µ , since L ξ a = dλ + [a, λ]. Therefore, for ξ = ∂ φ , the linear momentum on the cylinder readily integrates as Analogously, the variation of the energy is obtained for ξ = ∂ t , so that which by virtue of (2.6), integrates as expected, i.e., Note that, generically, there might be additional nontrivial solutions of eq. (3.11) that would lead to further conserved charges.
As a closing remark of this section, it must be emphasized that in order to construct an integrable system with the BMS 3 Poisson structure, one should at least specify the precise form of the Hamiltonian, so that the general solution of the consistency condition for the parameters in (3.11) could be obtained. Explicit examples of integrable systems of this sort that actually belong to an infinite hierarchy of them are discussed in the next section.

Hierarchy of integrable systems with BMS 3 Poisson structure
In this section we introduce a bi-Hamiltonian hierarchy of integrable systems with BMS 3 Poisson structure in a constructive way. We start from an extremely simple case, which nonetheless, possesses the key ingredients in order to propose a precise nontrivial integrable system of this type, that can be extended to an entire hierarchy labeled by a nonnegative integer k. The contact with some known results in the literature for certain particular cases is also addressed. Furthermore, a wide class of analytic solutions are explicitly constructed for an arbitrary representative of the hierarchy, including a couple of simple and interesting particular examples.
The field equations then explicitly reaḋ J = J + aP , which are trivially integrable. Indeed, the general solution of (4.3) on the cylinder is described by left movers and it can be expressed in terms of periodic functions M = M(t + φ) and N = N (t + φ), so that it reads Besides, since the field equations are very simple in this case, the consistency condition for the parameters of their symmetries in (3.11) becomes independent of the dynamical fields and their spatial derivatives, which explicitly reduces tȯ Note that (4.5) coincides with the field equations in (4.3) for ε J = P and ε P = J , and hence the general solution of the consistency conditions for the parameters is also given by chiral (left mover) functions as in (4.4). Therefore, the variation of the canonical generators in (3.12) readily integrates as The algebra of the conserved charges (4.6) can then be directly obtained from their corresponding Poisson brackets. As a shortcut, by virtue of the algebra can also be read from the transformation law of the fields in (3.8), and it is then found to be given precisely by the BMS 3 algebra in (2.4), which once expanded in modes reads as in eq. (1.1).
It is worth highlighting that this simple dynamical system actually turns out to be bi-Hamiltonian. This is so because the field equations can be expressed in terms of two different Poisson structures, so that (4.3) can be written as where D (2) is the BMS 3 one in (2.1), while D (1) stands for the "canonical" Poisson structure, defined through the following differential operator In (4.8) the functions µ and thus, according to (2.6), the canonical Poisson structure (4.9) is associated to the following Hamiltonian In sum, the analysis of this extremely simple dynamical system with BMS 3 Poisson structure D (2) , being trivially integrable, allows to unveil a naturally related Poisson structure given by D (1) . The presence of both Poisson structures turns out to be the key in order to proceed with the construction of nontrivial integrable systems as well as an entire hierarchy associated to them. Indeed, one can verify that any linear combination of D (1) and D (2) also defines a Poisson structure, which guarantees the existence of the type of hierarchy that we are searching for.
Furthermore, and remarkably, the simple dynamical system described in this section can be seen to be equivalent to the Einstein equations for the reduced phase space that is obtained from a suitable set of boundary conditions for General Relativity in three spacetime dimensions, including its extension with purely geometrical parity-odd terms in the action. This is discussed in section 5. It is also worth pointing out that our field equations (4.4), in the case of c J = a = 0, can be interpreted as the ones of a compressible Euler fluid [72].
In the next subsection we carry out the explicit construction and the analysis of a simple, but nontrivial, integrable system with BMS 3 Poisson structure.

Integrable bi-Hamiltonian system (k = 1)
The first nontrivial integrable system of our hierarchy is obtained from (2.7) with µ J = µ P are given by eq. (4.10), so that the Hamiltonian corresponds to H (1) in (4.11). The field equations are then explicitly given bẏ Note that P evolves according to the KdV equation, 2 while the remaining equation is linear in J , with an inhomogeneous source term that is entirely determined by P and their spatial derivatives.
The field equations in (4.12) can also be seen to arise from a bi-Hamiltonian system with the same BMS 3 and canonical Poisson structures given by (2.1) and (4.9), respectively. Indeed, they can be written as where the functions µ J and µ (2) P are given by which can be obtained from the functional derivatives of a different Hamiltonian, as in (2.6), that reads being clearly conserved.

Symmetries
As explained in section 3.1, in order to find the remaining conserved quantities, it is necessary to find the general solution of the consistency conditions in (3.11) for the functions ε J and ε P that parametrize the symmetries of the field equations. In this case (k = 1), the consistency conditions in (3.11), with µ J and µ P given by eq. (4.10), explicitly reduce tȯ Note that the equations in (4.16) are linear for the parameters ε J and ε P . However, finding their solution is not that simple because their coefficients depend on J and P, who evolve according to the nonlinear field equations in (4.12). Nevertheless, if one assumes that the parameters are given by local functions of the dynamical fields and their spatial derivatives, the general solution of the consistency conditions (4.16) is found to be given by a linear combination of two independent arrays, K (j) andK (j) , that stand for a suitable generalization of the Gelfand-Dikii polynomials. The solution can then be written as where η j andη j are arbitrary constants, and both generalized polynomials K (j) andK (j) fulfill the same recursive relationship, given by If the initial seeds of the independent arrays are chosen as the recursion relation (4.18) then implies that the remaining ones are given by where R (n) stand for the standard Gelfand-Dikii polynomials, while T (n) correspond to a different set of polynomials that fulfill the following recursion relationships Remarkably, both sets of polynomials can be obtained from the variation of two independent functionals, H (n) where H

(n)
KdV stands for n-th conserved quantity of the KdV equation, whileH (n) [P, J ] depends linearly on J and it is given bỹ Therefore, the generalized polynomials can also be compactly defined as An explicit list of the first six polynomials R (n) and T (n) , with their corresponding functionals H (n) KdV andH (n) is given in appendix A.

Conserved charges
Since the general form of the parameters that describe the symmetries of the field equations has been explicitly found to be given by (4.17), by virtue of the fact that the generalized polynomials come from the functional derivatives of suitable functionals as in (4.26), the variation of the canonical generators in (3.12) reduces to which then readily integrates as KdV +η jH (j) .
(4.28) Therefore, we have explicitly found two infinite independent towers of conserved quantities, being spanned by H where the subscripts for the brackets in (4.29) stand for the corresponding Poisson structures. 3 Note that the pair of Hamiltonians that yield the same field equations in (4.13), given by (4.11) and (4.15), can then be written as KdV . (4.30) In sum, as pointed out in the introduction, eq. (4.28) turns out to be an explicit realization of the infinite set of commuting conserved charges that is constructed out from precise nonlinear combinations of the generators of the BMS 3 algebra and their spatial derivatives. As discussed in section 4.3, this provides the basis to extend this integrable system to an entire hierarchy.

Remarks on some additional symmetries
Apart from the infinite set of symmetries spanned by (3.8), with ε J and ε P given by (4.17), the field equations (4.12) can also be seen to be invariant under Galilean and anisotropic scale transformations: Galilean transformations.-They are parametrized by a single constant velocity parameter v 0 , so that the coordinates and the fields transform according tō respectively. Anisotropic scaling of Lifshitz type.-This symmetry is defined through a constant parameter σ, and it is generated bȳ which corresponds to an anisotropic scaling of Lifshitz type with dynamical exponent z = 3 (for a deeper discussion on anisotropic scaling of Lifshitz type, see e.g. [68,[74][75][76][77][78][79][80][81][82]). For later purposes, it is worth noting that both sets of conserved quantities, H KdV andH (n) , scale under (4.33) according toH (4.34)

Equivalence with the Hirota-Satsuma coupled KdV system of type ix
Here we show that the field equations of the bi-Hamiltonian integrable system with BMS 3 and canonical Poisson structures, described by (4.12), can be seen to be equivalent to a particular class of a generalization of the Hirota-Satsuma coupled KdV system [83]. Specifically, according to the classification in [84], the equivalence is shown to hold for the field equations of type ix, which have been shown to be integrable through a method that differs from the one we have used above. The equivalence can be seen as follows.
If one changes the dynamical fields and rescale time according to the field equations in (4.12) read with γ ≡ c J + ac P . (4.38) The equations in (4.36), (4.37) then turn out to be precisely the ones of type ix in [84]. Thus, at the level of the field equations there are actually only two inequivalent cases. The generic one corresponds to γ = 1, since the field equation in (4.36) can always be brought to this form provided that v is rescaled as v → γv. The remaining case is described by γ = 0, which is also known in the literature as "perturbed KdV" (see e.g. [85], [86], [87], [88], [89], [90], [91]). Note that for γ = 0 (c J = −ac P ), configurations with v = 0 (J = −aP) are devoid of energy, since H = H (0) in (4.2) vanishes. Nonetheless, they are generically endowed with both towers of conserved charges H (j) KdV andH (j) .
In sum, we would like to emphasize that both cases (γ = 0, 1), have now been shown to be endowed with radically inequivalent bi-Hamiltonian structures depending on whether the BMS 3 central charge c J vanishes or not. Therefore, these structures provide all what is needed in order to generalize the integrable system discussed in this subsection to a hierarchy of them that is labeled by a nonnegative integer k.

The hierarchy (k ≥ 0)
The results obtained in section 4.2 ensure that a hierarchy of bi-Hamiltonian integrable systems with BMS 3 and canonical Poisson structures, given by D (2) and D (1) , can be readily constructed out from choosing any of their Hamiltonians to be given by an arbitrary linear combination of the independent conserved charges H (n) KdV andH (n) in (4.25). Hereafter we focus in the subclass of them that possesses well-defined scaling properties. In order to achieve this task, we recall that if one rescales the spacelike coordinate and the fields as in (4.33), then both sets of conserved charges scale according to (4.34). Therefore, the most general combination that possesses the suitable scaling properties that we look for is described by a Hamiltonian of the form with a constant and any fixed value of the integer k ≥ 0. The field equations of the corresponding hierarchy of integrable systems then read The hierarchy defined through (4.40) is clearly bi-Hamiltonian, since by virtue of the recursion relationship in (4.18), the field equations can also be expressed as Furthermore, by construction, the field equations for any representative of the hierarchy turn out to be invariant under anisotropic scaling transformations given by which is of Lifshitz type, and characterized by a dynamical exponent z = 2k + 1.
The field equations of the hierarchy can also be explicitly written in terms of the polynomials T (k) defined through (4.24) and the Gelfand-Dikii polynomials R (k) , so that they readJ (4.44) It is worth pointing out that for any representative of the hierarchy with k > 1, not only the field equations in (4.44), but also the consistency condition for the functions ε J and ε P that parametrize the symmetries in (3.11) become severely more complicated than the simplest cases of k = 0, 1 (see eqs. (4.3), (4.5), and (4.12), (4.16), respectively). However, and remarkably, when ε J and ε P are assumed to be given by local functions of the dynamical fields and their spatial derivatives, the general solution of the consistency condition for the parameters in (3.11) for k ≥ 1 turns out to be precisely given by the same expansion in terms of the generalized Gelfand-Dikii polynomials K (j) andK (j) , as in (4.17) for k = 1. Therefore, as a consequence, the corresponding canonical generators turn out to be given by the two independent sets of conserved charges given by (4.28), which are in involution, i.e., the commuting charges fulfill (4.29) for both Poisson structures. Nevertheless, depending on the choice of Poisson structure, D (2) or D (1) , the energy of the system in (3.15), now corresponds to the Hamiltonian, H (k) or H (k+1) , defined through (4.39), respectively.
As an ending remark of this subsection, it is worth noting that the field equations of the so-called "perturbed KdV hierarchy" (see e.g., [85,87]) are precisely recovered from (4.44) for the particular case of c J = a = 0. In this case, for the entire hierarchy, according to (4.39), configurations with J = 0 do not carry energy, which goes by hand with the fact that the Hamiltonian corresponds to the energy of a perturbation described by J . Indeed, for this class of configurations, the entire set of conserved chargesH (j) in (4.25) also vanishes. Note that the remaining ones, given by H

(j)
KdV generically remain nontrivial, which can be interpreted as the conserved charges associated to an arbitrary "background configuration" described by P = P (t, φ) that solves the field equations of the k-th representative of the KdV hierarchy.

Analytic solutions
Exact analytic solutions of the KdV equation, as well as for the k-th representative of the KdV hierarchy, have been thoroughly studied in the literature since long ago through different methods (see e.g., [69,92]).
Here we show how to obtain an interesting wide class of analytic solutions of the field equations in (4.44) for an arbitrary value of the nonnegative integer k that labels our hierarchy of integrable systems with BMS 3 Poisson structure.
As a warming up exercise, let us begin considering the simplest case, described by choosing the field P to be constant, so that the field equation of the KdV hierarchy in (4.44) is trivially solved for an arbitrary value of k. In this case, the field equation for J (t, φ) in (4.44) just reduces to a dispersive linear homogeneous equation with constant coefficients, given byJ with α k,m ≡ (2k+1)!! m!(2k−2m+1)!! , which can be easily solved for an arbitrary member of the hierarchy. Indeed, expanding in Fourier modes according to 4 one finds that the corresponding dispersion relation is given by In the case of nontrivial solutions P = P(t, φ) of the k-th KdV equation, it is also possible to find generic analytic solutions for J (t, φ). Remarkably, in spite of the fact that J (t, φ) obeys a linear differential equation, their exact solutions are able to be nondispersive. This effect occurs because the coefficients of the linear equation for J in (4.44) are determined by nontrivial solutions of the k-th KdV equation, and it persists even in presence of a source term (with a = 0 or c J = 0). This is explicitly discussed in what follows.

Generic analytic solutions
Let us assume that P = P(t, φ) corresponds to an arbitrary generic solution of the field equations of the k-th representative of the KdV hierarchy, described by the second line of (4.44). Note that, since the central extension c P has not been scaled away, nontrivial solutions P(t, φ) explicitly depend on c P .
In order to find the form of J = J (t, φ) one has to solve the remaining equation in (4.44), which turns out to be linear in J , and possesses an inhomogeneous term that is completely specified by P and their spatial derivatives. Hence, the generic solution for J is given by the sum of the particular and the homogeneous solutions, i.e., Noteworthy, as it is shown in appendix C, for an arbitrary value of the label k of the hierarchy, a particular solution J = J p (t, φ) can be analytically expressed in a very compact way, so that it reads 5 Besides, a generic solution of the homogeneous equation can be found by virtue of the symmetries in (3.8), being spanned by the subset of parameters given by (4.17) that preserve the form of P(t, φ), i.e., the ones for which δP = 0. The suitable subset of symmetries we look for then becomes generated by an arbitrary combination of the generalized polynomials K (j) , excludingK (j) ; and hence the parameters are given by ε J and ε P in (4.17) withη j = 0. Thus, according to (3.8), the homogeneous solution is given by Therefore, by virtue of the recursive relation of the Gelfand-Dikii polynomials in (4.21), the generic solution for J acquires the form As a cross-check, in appendix C it is explicitly shown that eq. (4.51) solves the field equation for J in (4.44).
In sum, the generic solution for J in (4.51) has been generated through acting on the particular solution J p with the symmetries that are spanned by the corresponding canonical generators in (4.28) withη j = 0. Hence, since the generators are in involution, the full set of conserved charges for the solution characterized by the fields P and J p must coincide with the ones for the fields P with J given by (4.51). In other words, the homogeneous part of the solution J h does not contribute to the conserved charges.
The conserved charges are then described by the corresponding ones for KdV, given by H

(n)
KdV , together with the independent setH (n) in (4.25). Once the latter set is evaluated in the generic solution (4.51), it can be compactly written in terms of the total derivative of H

(n)
KdV with respect to c P , so that it reads (see appendix C) Therefore, for the BMS 3 Poisson structure D (2) , the energy of our generic solution for the k-th representative of the hierarchy is given by the Hamiltonian in (4.39), which reduces to It is worth pointing out that the conserved charges of the solution can be expressed exclusively in terms of P and their spatial derivatives.
Note that the generic class of solutions presented here is mapped into itself under the anisotropic scaling of Lifshitz type given in (4.43), where the arbitrary constants η j of the homogeneous solution (4.50) transform asη j = σ 2j−1 η j .
Additionally, as pointed in section 4.2.3, in the case of k = 1 the field equations are also invariant under Galilean transformations. Hence, in this case, by virtue of the Galilean boost spanned by (4.31), our solution in (4.51) acquires nontrivial zero modes once expressed in the moving frame.
In the next subsection, we explicitly describe a couple of simple and interesting particular examples of analytic solutions in the case of k = 1.

Particular cases for k = 1
Single KdV soliton on the real line.-The integrable system described in section 4.2 can be extended to R 2 provided that the angular coordinate is unwrapped (−∞ < φ < ∞), so that our previous analysis still holds once the fall-off of the fields is assumed to be fast enough so as to get rid of boundary terms.
In our conventions, the well-known single soliton solution of the KdV equation in (4.12) reads , and v stands for the integration constant that parametrizes the velocity and amplitude of the soliton.
An analytic solution for the remaining field equation in (4.12) can then be constructed out from (4.51), with P given by (4.54). For simplicity we consider that the integration constants in (4.51) are chosen as η j = η 1 δ j,1 , with η 1 arbitrary, so that the solution for J becomes explicitly given by Therefore, although J obeys a linear differential equation, the solution in (4.55) clearly maintains its shape as it evolves in time. The profile of J in (4.55) is depicted in figure 1.
Cnoidal wave on S 1 .-In the case of periodic boundary conditions (−π ≤ φ < π) an analytic solution for the KdV equation in (4.12) of solitonic type is known as a "cnoidal wave", since it is described in terms the Jacobi elliptic cosine (cn) (see e.g. [92]). The solution is given by with y = α m (φ − c P vt), and the velocity parameter is related to the remaining integration constants as The wavelength of the solution is given by 2 m α K(m), where K stands for a complete elliptic integral of the first kind. Accordingly, the elliptic parameter m can take values within the range 0 < m < 1, satisfying 2 m α K(m) = 2π n with n ∈ N. As in the previous example, we then construct the analytic solution for J from the generic one in (4.51), with P given by the cnoidal wave in (4.56). For the sake of simplicity, we again choose the integration constants to be given by η j = η 1 δ j,1 , and hence the searched  Note that the profile of J preserves its form as it evolves in time only in the case of γ = ac P + c J = 0 (perturbed KdV), otherwise the amplitude grows linearly with time. Nonetheless, the energy in (4.53) as well as the remaining conserved charges given by H As an ending remark of this section, it is worth pointing out that in the special case of c J = a = 0, the field equation for J becomes devoid of a source. The particular solutions, given by (4.55) for the real line, and by (4.58) in the case of S 1 , in this case turn out to be described by odd analytic functions that maintain its shape as they propagate with the same velocity as their corresponding KdV solitons described by P in (4.54) and (4.56), respectively. Note that, according to eq. (4.53), the total energy of this sort of solitonantisoliton bound states for J vanishes, and the conserved chargesH (n) in (4.25) also do. Nevertheless, the conserved charges H  5 Geometrization of the hierarchy: the dynamics of locally flat spacetimes in 3D In this section, we show that the entire structure of the class of integrable systems with BMS 3 Poisson structure described above can be fully geometrized, in the sense that the dynamics turns out to be equivalently understood through the evolution of spacelike surfaces embedded in locally flat spacetimes in three dimensions. For the sake of simplicity, here we focus in the case of BMS 3 Poisson structures with c J = 0. Thus, following the lines of [68], it is possible to unveil a deep link between the class of integrable systems aforementioned and General Relativity in three spacetime dimensions. Concretely, here we show that the Einstein-Hilbert action without cosmological constant in 3D can be endowed with an appropriate set of boundary conditions, so that in the reduced phase space, the Einstein equations in vacuum, which imply the vanishing of the Riemann tensor, precisely reduce to the ones of the dynamical systems with BMS 3 Poisson structure in (2.7). As a consequence, it is possible to establish a one-to-one map between any solution of this kind of integrable systems and certain specific locally flat metric in three spacetime dimensions. Furthermore, the symmetries of the integrable systems can be seen to naturally emerge from diffeomorphisms that preserve the asymptotic form of the spacetime metric. Hence, and remarkably, the symmetries manifestly become Noetherian in our geometric framework. Therefore, the infinite set of conserved charges for the integrable system is transparently recovered from the corresponding surface integrals in the canonical approach. This can be seen as follows.
The Einstein-Hilbert action in three spacetime dimensions can be equivalently expressed as a Chern-Simons action for the isl(2, R) algebra [93], [94]. Thus, up to boundary terms, the action (5.1) can be written as where · · · stands for the invariant bilinear form defined in eq. (3.2) with c J = 0 and c P = 1/ (8πG). The components of the isl(2, R)-valued gauge field are then identified with the dualized spin connection and the dreibein according to In order to describe the asymptotic structure of the fields, as explained in [95], [96], [97] it is useful to choose the gauge so that the connection reads where the radial dependence is completely captured by the group element b = b (r), which as shown in [62] can be conveniently chosen as b = e rP 2 . One of the advantages of this gauge choice is that the remaining analysis can be performed in terms of the auxiliary connection a = a t dt + a φ dφ , (5.5) that only depends on t, φ. Here we propose that the asymptotic form of the auxiliary connection for the gravitational field in (5.5) is precisely given by the two-dimensional locally flat gauge field that describes the field equations of the dynamical system with BMS 3 Poisson structure in section 3. Their components a φ and a t are then described by eqs. (3.3) and (3.4), respectively. Therefore, from (5.2), the field equations imply that the connection A is flat (F = dA + A 2 = 0), which by virtue of (5.3) amounts to deal with three-dimensional manifolds with vanishing curvature and torsion; whereas eq. (5.4) means that the field strength of the auxiliary gauge field also vanishes as in (3.6). Hence, for our boundary conditions, the Einstein equations in vacuum precisely reduce to the ones of a dynamical system with BMS 3 Poisson structure in (2.5). Besides, the asymptotic symmetries, being defined as the diffeomorphisms that preserve the asymptotic form of the spacetime metric, turn out to be equivalent to the set of gauge transformations δA = dλ + [A,λ] that maintain the asymptotic form of the gauge field in (5.4). For our boundary conditions, one then finds that the asymptotic symmetries are spanned by a Lie-algebra-valued parameter of the formλ = b −1 λb, where λ = Λ (ε J , ε P ) is exactly given as in eq. (3.7), including the consistency condition for the parameters ε J , ε P in (3.11). Furthermore, the transformation law of the dynamical fields J , P precisely agrees with the transformations in eq. (3.8).
Since the asymptotic symmetries are Noetherian, the global conserved charges can be readily obtained using the canonical approach [98]. Indeed, their variation is explicitly given by surface integrals defined at the boundary of the spatial section, 6 which read and precisely coincide with the variation of the conserved charges introduced in eq. (3.12), in the context of integrable systems. In particular, if the Lagrange multipliers µ J and µ P are kept fixed at the boundary according to (4.41), so that they correspond to the variation of the functional H (k) defined in (4.39), the Einstein equations reduce to the ones of the hierarchy of integrable systems discussed in section 4.3. Therefore, in this case, the variation of the surface integrals in (5.6) integrates precisely as in (4.28). Note that the energy of a gravitational configuration that fulfills these boundary conditions is then given by the Hamiltonian of the corresponding integrable system as in (3.15) In the simplest case of k = 0, described in section 4.1, we recover the set of boundary conditions proposed in [62] 7 (see also [61]) which contain the boundary conditions in [5] for a particular choice of Lagrange multipliers at the boundary. Note that in this case, the BMS 3 algebra is realized as the asymptotic symmetry algebra. For the remaining cases (k ≥ 1), the new class of boundary conditions is such that the asymptotic symmetry algebra is infinite-dimensional, abelian and devoid of central charges, which is equivalent to the fact that the conserved charges of the hierarchy are in involution (see eq. (4.29)).
It is worth pointing out that, in the metric formalism, our particular choices for the Lagrange multipliers µ J and µ P correspond to fixing the lapse and shift functions in the ADM decomposition of the metric, as suitable local functionals of the dynamical variables at the boundary. Furthermore, and remarkably, the class of locally flat spacetimes described by this new set of asymptotic conditions inherits the anisotropic Lifshitz scaling of the corresponding integrable systems by virtue of the boundary conditions. This effect then becomes the flat analogue of the one found in [68] for the case of locally AdS 3 spacetimes, where isotropy is recovered in the particular case of the Brown-Henneaux boundary conditions [99].
It is also amusing to verify that trivial solutions of the class of integrable systems described above, like the ones just given by configurations with J and P constants, actually correspond to the geometries of the flat cosmological spacetimes in [100]. Hence, from a gravitational point of view, these configurations become certainly non-trivial, not only in a geometric sense, but also in the physical one since they carry Hawking temperature and entropy associated to the cosmological horizon. It would then be worth to explore additional simple configurations for the integrable systems that could naturally become non-trivial in the gravitational framework and vice versa.
The precise details of the asymptotic structure in the metric formulation, including the study of some of the physical properties of the flat cosmological spacetimes once they are embedded within the new set of boundary conditions presented in this section will be discussed in a forthcoming work. It can also be shown that the geometrization of the class of integrable systems discussed here, but in the generic case with c J = 0, can be performed through a suitable extension of the analysis in [7], [65] once suitable parity odd terms in the action are included (see also [101]).

Extensions of our results
As pointed out in the introduction, the BMS 3 algebra admits some extensions which, according to our results, might be expected to be linked to new classes of integrable systems. In particular, an interesting nonlinear extension of BMS 3 that includes additional generators of spin s > 2 was found in [60] (in full agreement with the algebra simultaneously found in [59] for s = 3). This kind of extensions can be regarded as "flat W -algebras", since they can be recovered from a suitable Inönü-Wigner contraction of two copies of certain classical W -algebras (for a review about W -algebras, see e.g. [102]). Noteworthy, preliminary results [103] show that new hierarchies of integrable systems whose Poisson structures correspond to flat W -algebras indeed exist. In fact, this family of integrable systems turns out to be bi-Hamiltonian, and furthermore, following the lines of section 5, they can also be geometrized in terms of higher spin gravity without cosmological constant in three spacetime dimensions endowed with a suitable set of boundary conditions. Therefore, the symmetries of this novel class of integrable systems can be seen as combinations of diffeomorphisms and higher spin gauge transformations that preserve the asymptotic form of the three-dimensional configurations. As a consequence, in the three-dimensional geometric setup, the infinite set of conserved charges of the integrable systems emerge as the canonical generators that correspond to the asymptotic symmetries, being described by suitable surface integrals at the boundary, which turn out to be in involution.

B Involution of the conserved quantities
In order to prove that our set of conserved charges, H (m) = H (n) KdV ;H (n) , is abelian in both Poisson brackets, one can follow the lines of the proof of the same statement in the case of the pure KdV equation that can be found in standard textbooks (see e.g., [69,73]).
Without loss of generality, let us assume that m > n. Thus, the Poisson bracket associated to the BMS 3 operator D (2) of two conserved charges, which reads T (k) = c J ∂R (k) ∂c P +ˆdϕJ (ϕ) δR (k) (ϕ) δP . (C.5) Thus, once (C.5) is evaluated on (C.1), it reduces to η jˆd ϕ∂ ϕ R (j+1) (ϕ) δR (k) δP(ϕ) + c J dR (k) dc P + at∂ t R (k) , (C. 6) where dR (k) dc P stands for the total derivative of R (k) with respect to the central charge c P , given by dR (k) dc P = ∂R (k) ∂c P +ˆdϕ ∂P(ϕ) ∂c P δR (k) δP(ϕ) . (C.7) Besides, the time derivative of (C.1) can be written aṡ where we have made use of the k-th KdV equation in (C.3), as well as the recursion relation for the Gelfand-Dikii polynomials in (4.21). Note that by virtue of (C.4), the following identity holdŝ (C.9) where the first term in the r.h.s. of (C.9) vanishes due to the fact that the conserved charges H