# Extensions of dark KdV equations: nonhomogeneous classifications,
bosonizations of fermionic systems and supersymmetric dark systems

S. Y. Lou
School of Physical Science and Technology, Ningbo University, Ningbo, 315211,
China

###### Abstract

Dark equations are defined as some kinds of integrable couplings with some
fields being homogeneously and linearly coupled to others. In this paper, dark
equations are extended in several aspects. Taking the Korteweg-de Vrise (KdV)
equation as an example, the dark KdV systems are extended to nonhomogenous
forms, nonlinear couplings and graded linear cases. The two-component
nonhomogeneous linear coupled dark KdV systems are completely classified. The
nonlinear coupled dark KdV systems may be obtained through the decompositions
from higher dimensional integrable systems like the B-type KP equation. Graded
linear coupled dark KdV systems may be produced by introducing dark parameters
(including the Grassmann parameters) to usual integrable systems. Especially,
applying the bosonization approach to the integrable systems with fermion
fields such as the supersymmetric integrable systems and super-integrable
models, infinitely many graded linear dark systems can be generated. Finally,
the dark KdV systems are extended to supersymmetric ones. The full
classifications for the supersymmetric dark KdV systems are obtained related
to two types of usual supersymmetric KdV equations.

##  1 Introduction

Astrophysical investigations have yielded findings indicating that the
observed motion of stars cannot be adequately explained without postulating
the existence of a significant amount of mass in the Universe attributed to a
hypothetical entity known as dark matter (DM) [1, 2]. Additionally, the
concept of dark energy (DE) has been introduced to account for the observed
accelerating expansion of our Universe [3, 4, 5]. In a notable work by
Kupershmidt [6], a distinct concept termed “dark equations” is proposed, which
systematically unveils equations possessing unconventional properties, some of
which may be characterized as strange and mysterious. The dark equations, as
defined and discovered by Kupershmidt, exhibit similarities with linearized
representations of existing systems or the study of the interaction between an
external linear wave and a particular solution of an integrable system.

In this paper, we present an expansion of the dark equation concept across
multiple aspects, encompassing nonhomogeneous linear extensions, graded linear
extensions, and nonlinear extensions. For the KdV type of dark systems,
Kupershmidt’s work has yielded a comprehensive classification under the
condition of homogeneous linearity with only one single dark field. In section
2, we revisit the classification of dark KdV systems within the context of
nonhomogeneous linearity, aiming to uncover additional dark KdV systems with
multiple dark field components. To achieve this, we extend the dark systems to
the nonlinear case. The first type of nonlinear extensions is achieved by
introducing dark parameters into conventional integrable systems such as the
KdV equation. Additionally, these nonlinear extensions can be subjected to
graded linearization for the dark fields. In section 3, we investigate the
dark KdV systems by introducing a single dark parameter. The dark KdV systems
obtained through the introduction of a single dark parameter can also be
derived from the perturbation expansion of the standard KdV equation, as
discussed in [7].

It is well-established in scientific literature that supersymmetry plays a
significant role in addressing crucial physical phenomena, including gravity
[8] and dark matter [9, 10, 11]. Notably, a supersymmetric integrable model
possesses the intriguing property of encompassing an infinite number of dark
equations with an infinite number of dark fields. Sections 4 and 5 of this
study employ the bosonization method, as proposed in Ref. [12], to derive
various integrable graded linear dark KdV systems from certain supersymmetric
KdV systems. Remarkably, the bosonization method can be applied to solve
diverse types of integrable systems, encompassing fermion fields as well. In
section 6, the bosonization procedure is utilized to obtain distinct
integrable graded linear dark KdV systems from the super-integrable KdV
equation [13]. Section 7 focuses on extending supersymmetric KdV systems to
supersymmetric dark KdV systems. These supersymmetric dark KdV systems can be
effectively bosonized, leading to the generation of additional integrable
graded linear KdV systems. Section 8 highlights the existence of genuinely
nonlinear integrable dark KdV systems. Notably, one exceptional nonlinear
integrable dark KdV system arises from the special Bäcklund transformation of
the B-type Kadomtsev-Petviashvilli equation, employing the decomposition
approach [14]. Section 9 is dedicated to extending conventional dark KdV
systems to supersymmetric dark KdV systems. Finally, the last section presents
concluding remarks.

##  2 Classification of nonhomogeneous linear dark KdV systems

###  2.1 Definitions of dark equations

A given nonlinear partial differential equation (PDE),

$u_{t}=X(u),\quad u=(u_{1},\ u_{2},\ \ldots,\ u_{m})^{T},$ (1)

where the superscript $T$ means the transposition of a matrix, is called
symmetry integrable if there is a higher order symmetry flow system

$u_{\tau}=Y(u),$ (2)

such that the consistent commuting condition,

$u_{t\tau}-u_{\tau t}=[X,\ Y]=X^{\prime}Y-Y^{\prime}X=0,$ (3)

is identically satisfied with $X^{\prime}$ and $Y^{\prime}$ being the
linearized operators of $X$ and $Y$, respectively.
Definition 1. Homogeneous linear dark systems. An integrable dark system [6]
is defined as an extension of (1) and (2) in the forms

$U_{t}=\left(\begin{array}[]{c}u\\\
v\end{array}\right)_{t}=\tilde{X}(U)=\left(\begin{array}[]{c}X(u)\\\
A(u)v\end{array}\right),$ (4)

and

$U_{\tau}=\left(\begin{array}[]{c}u\\\
v\end{array}\right)_{\tau}=\tilde{Y}(U)=\left(\begin{array}[]{c}Y(u)\\\
B(u)v\end{array}\right),$ (5)

where $U\equiv(u,\ v)^{T}=(u_{1},\ u_{2},\ \ldots,\ u_{m},\ v_{1},\ v_{2},\
\ldots,\ v_{n})^{T}$, if the compatibility condition

$U_{t\tau}-U_{\tau t}=[\tilde{X},\
\tilde{Y}]=\tilde{X}^{\prime}\tilde{Y}-\tilde{Y}^{\prime}\tilde{X}=0,$ (6)

is still identically satisfied. $A=A(u)$ and $B=B(u)$ in (4) and (5) are
$n\times n$ matrix operators depending only on the field $u$.

The dark system described by equation (4) elucidates the interaction between
the dark fields $v$ and a specific solution $u$ of the given system (1).
According to Kupershmidt’s definition (4), it is established that the
equations governing the dark fields $v$ are linear and homogeneous (with a
trivial vacuum solution $v=0$). Consequently, these types of dark systems are
referred to as homogeneous linear dark systems.
Definition 2. Nonhomogeneous linear dark systems. An integrable nonhomogeneous
linear dark system

$U_{t}=\left(\begin{array}[]{c}u\\\
v\end{array}\right)_{t}=\bar{X}(U)=\left(\begin{array}[]{c}X(u)\\\
A(u)v+A_{0}(u)\end{array}\right),$ (7)

is defined as if it is consistent with

$U_{\tau}=\left(\begin{array}[]{c}u\\\
v\end{array}\right)_{\tau}=\bar{Y}(U)=\left(\begin{array}[]{c}Y(u)\\\
B(u)v+B_{0}(u)\end{array}\right),$ (8)

i. e., $U_{t\tau}-U_{\tau t}=[\bar{X},\ \bar{Y}]=0$ is identically satisfied
for the suitable operators $\\{A(u),\ B(u)\\}$ and the nonzero vectors
$\\{A_{0}(u),\ B_{0}(u)\\}$.

Different from (4), $v=0$ is not a solution of (7) except for some special
solutions $u=\bar{u}$ with $\\{\bar{u}_{t}=X(\bar{u}),\ A_{0}(\bar{u})=0\\}$.
In fact, the dark system (4) can be further extended to nonlinear coupled
forms for the dark field $v$.
Definition 3. Nonlinear dark systems. An integrable nonlinear dark system

$U_{t}=\left(\begin{array}[]{c}u\\\
v\end{array}\right)_{t}=\hat{X}(U)=\left(\begin{array}[]{c}X(u)\\\ A(u,\
v)\end{array}\right),$ (9)

can be defined as if it possesses a higher order symmetry flow

$U_{\tau}=\left(\begin{array}[]{c}u\\\
v\end{array}\right)_{\tau}=\hat{Y}(U)=\left(\begin{array}[]{c}Y(u)\\\ B(u,\
v)\end{array}\right),$ (10)

such that $U_{t\tau}-U_{\tau t}=[\hat{X},\ \hat{Y}]=0$.

In the definition of the nonlinear dark system (9), at least one of the
components of the vector $A\equiv A(u,\ v)\equiv\\{A_{1},\ A_{2},\ \ldots,\
A_{n}\\}$ is nonlinear for some components of $v=\\{v_{1},\ v_{2},\ \ldots,\
v_{n}\\}$.
Definition 4. Graded linear dark systems. A special nonlinear dark system (9)
is defined as a graded linear dark system if

$A_{i}=\hat{A}_{i}(u,\ v_{1},\ v_{2},\ \ldots,\ v_{i-1})v_{i}+\alpha_{i}(u,\
v_{1},\ v_{2},\ \ldots,\ v_{i-1}),\ i=1,\ 2,\ \ldots,\ n,$ (11)

where the operators $\hat{A}_{i}$ and the functions $\alpha_{i}$ are $v_{k}$
independent for all $k\geq i$.

In the following section, we will begin by presenting a list of nonhomogeneous
linear dark KdV systems before providing specific examples of graded linear
dark systems.

##  3 Classification of nonhomogeneous linear dark KdV systems with one dark
field

For the KdV equation

$u_{t}=(3u^{2}-u_{xx})_{x},$ (12)

there are numerous dark KdV systems including nine types of the following dark
KdV systems with only one homogenous linear coupling [6] in the form

$\left\\{\begin{array}[]{l}u_{t}=(3u^{2}-u_{xx})_{x}\equiv J_{1x},\
J_{1}\equiv 3u^{2}-u_{xx},\\\ v_{t}=\alpha v_{xxx}+\beta uv_{x}+\gamma
u_{x}v,\end{array}\right.$ (13)

where $\alpha,\ \beta$ and $\gamma$ are arbitrary constants.

In this section, we just list the classification results on the nonhomogeneous
linear extensions of (13) in the form

$\left\\{\begin{array}[]{l}u_{t}=J_{1x},\\\ v_{t}=\alpha v_{xxx}+\beta
uv_{x}+\gamma
u_{x}v+\delta_{1}u_{xxx}+\delta_{2}u_{xx}+\kappa_{1}uu_{x}+\kappa_{2}u^{2},\end{array}\right.$
(14)

with possible constants $\alpha,\ \beta,\ \gamma,\ \delta_{1},\ \delta_{2},\
\kappa_{1}$ and $\kappa_{2}$.
Type 1. Lax pair dependent dark KdV system. It is known that the KdV equation
(12) possesses the strong Lax pair,

$\displaystyle\hat{L}\psi=0,\qquad\hat{L}\equiv\partial_{x}^{2}-u-\lambda,$
(15)
$\displaystyle\hat{S}\psi=0,\qquad\hat{S}\equiv\partial_{t}+4\partial_{x}^{3}-6u\partial_{x}-3u_{x}.$
(16)

The compatibility condition of (15) and (16)

$[\hat{L},\ \hat{S}]=\hat{L}\hat{S}-\hat{S}\hat{L}=0$

is nothing but the KdV equation (12).

The first two-component dark KdV system is the integrable coupling of the KdV
equation (12) and the time part of the Lax pair (16) with a nonhomogeneous
term, i.e.,

$\left\\{\begin{array}[]{l}u_{t}=J_{1x},\\\
v_{t}=-4v_{xxx}+6uv_{x}+3u_{x}v+a(uu_{x}-u_{xxx}),\end{array}\right.$ (17)

where $a$ is an arbitrary constant. When $a=0$, (17) is just one of the dark
equation given by Kupershmidt [6].

It is inherent to the nature of the dark KdV system, as represented by
equation (17), that its higher order symmetry flows can be understood as
integrable couplings involving the higher order KdV equations and their
respective time components of the Lax pairs, incorporating appropriate
nonhomogeneous terms.

To illustrate, one particular example is the fifth order flow derived from
equation (17)

$\left\\{\begin{array}[]{lll}u_{\tau}&=&J_{2x},\ J_{2}\equiv
10uu_{xx}+5u_{x}^{2}-10u^{3}-u_{xxxx},\\\
v_{\tau}&=&40uv_{xxx}-16v_{xxxxx}+60u_{x}v_{xx}+10(5u_{xx}-3u^{2})v_{x}+15(u_{xx}-u^{2})_{x}v\\\
&&+5a(3uu_{xx}-u_{xxxx})_{x}+5au_{x}(3u_{xx}-2u^{2}).\end{array}\right.$ (18)

By performing a direct analysis, it can be rigorously established that the
equations $u_{t\tau}=u_{\tau t}$ and $v_{t\tau}=v_{\tau t}$ are identically
satisfied, thereby confirming their validity without exception.
Type 2. Symmetry related dark KdV system. A symmetry, $\sigma$, of the KdV
equation (12) is defined as a solution of its linearized equation

$\sigma_{t}=(6u\sigma-\sigma_{xx})_{x}\equiv\hat{K}\sigma,\qquad\hat{K}\equiv\partial_{x}(6u-\partial_{x}^{2}),$
(19)

which means (12) is form invariant under the transformation $u\rightarrow
u+\epsilon\sigma$ with the infinitesimal parameter $\epsilon$.

The KdV equation (12) is known to possess a recursion operator denoted as
$\Phi$, $\Phi\equiv\partial_{x}^{2}-4u-2u_{x}\partial_{x}^{-1}$. This operator
has the remarkable property of transforming a known symmetry, $\sigma_{0}$,
into a new symmetry, denoted as $\sigma_{1}=\Phi\sigma_{0}$. Notably, the
eigenvalue problem associated with the recursion operator and the symmetry
equation

$\displaystyle\Phi\sigma=\lambda\sigma,\qquad\Phi\equiv\partial_{x}^{2}-4u-2u_{x}\partial_{x}^{-1},$
(20)
$\displaystyle\sigma_{t}=\hat{K}\sigma,\qquad\hat{K}\equiv\partial_{x}(6u-\partial_{x}^{2}),$
(21)

form the second type of Lax pair for the KdV equation (12). Consequently, it
is not surprising that the KdV equation (12), together with its symmetry
equation (21) accompanied by two nonhomogeneous terms, constitutes the second
type of integrable dark KdV systems

$\left\\{\begin{array}[]{l}u_{t}=J_{1x},\\\ v_{t}=(6uv-
v_{xx})_{x}-au_{xxx}+6buu_{x},\end{array}\right.$ (22)

where $a$ and $b$ are two arbitrary constants. One of the dark equations given
in [6] is related to (22) by $a=b=0$.

It is reasonable that the higher order symmetry flows of (22) are just the
higher order KdV equations coupled with their symmetry equations with some
suitable nonhomogeneous complements. For instance, the fifth order flow of
(22) possesses the form

$\left\\{\begin{array}[]{l}u_{\tau}=J_{2x},\\\
v_{\tau}=[10u_{xx}v+10(uv_{x})_{x}-30u^{2}v-v_{xxxx}]_{x}+cJ_{2x}+(a-b)(10u^{3}-u_{xxxx})_{x},\end{array}\right.$
(23)

with a further arbitrary constant $c$.
Type 3. Dark KdV system related to the dual symmetry. A dual symmetry,
$\tilde{\sigma}$, of the KdV equation (12) is defined as a solution of the
following linear equation

$\tilde{\sigma}_{t}=\tilde{K}\tilde{\sigma},\qquad\tilde{K}\equiv(6u-\partial_{x}^{2})\partial_{x},$
(24)

where $\tilde{K}$ is the dual operator of $-\hat{K}$ with $\hat{K}$ defined in
the symmetry equation (19).

The third type of dark KdV systems is a coupled system for the KdV equation
(12) and the dual symmetry equation (24) with two nonhomogeneous extensions,

$\left\\{\begin{array}[]{l}u_{t}=J_{1x},\\\
v_{t}=\tilde{K}v-au_{xx}+3bu^{2},\end{array}\right.$ (25)

where $a$ and $b$ are two arbitrary constants. When $a=b=0$, the dark KdV
system (25) is reduced back to that of Ref. [6].

The higher order symmetry flows associated with the third type of dark system
(25) can be expressed as a set of coupled equations. These equations consist
of the higher order KdV equations and their dual symmetry equations, augmented
with some nonhomogeneous extensions. Specifically, the fifth order symmetry
flow of the dark KdV system (25) is given by the following system of
equations:

$\left\\{\begin{array}[]{l}u_{\tau}=J_{2x},\\\
v_{\tau}=(10uv_{xx}-v_{xxxx})_{x}+10(u_{xx}-3u^{2})v_{x}+cJ_{2}+(a-b)(10u^{3}-u_{xxxx}),\end{array}\right.$
(26)

where the coefficients $a$, $b$, and $c$ are constants that determine the
specific characteristics of the system.
Type 4. Strange Dark KdV system [6] related to the supersymmetric KdV
equation. The fourth type of the dark KdV systems possesses the form

$\left\\{\begin{array}[]{l}u_{t}=J_{1x},\\\
v_{t}=\hat{Y}v+3a(u^{2}-u_{xx}),\qquad\hat{Y}\equiv\partial_{x}(3u-\partial_{x}^{2})\end{array}\right.$
(27)

with an arbitrary constant $a$.

The integrability of (27) is guaranteed by the existence of higher order
symmetries. The fifth order symmetry flow of (27) reads,

$\left\\{\begin{array}[]{l}u_{\tau}=J_{2x},\\\
v_{\tau}=[5uv_{xx}+5(u_{x}v)_{x}-10u^{2}v-v_{xxxx}]_{x}+5a(5uu_{xx}+4u_{x}^{2}-2u^{3}-u_{xxxx}).\end{array}\right.$
(28)

The special case of (27) with $a=0$ has been given in Ref. [6].

In Section 5 of this study, we will explore the connection between the dark
KdV system, represented by equation (27), and the integrable supersymmetric
KdV equation. We will analyze and discuss the relationship between these two
equations, highlighting their similarities, differences, and potential
implications. This investigation aims to deepen our understanding of the
underlying connections and uncover any significant insights that arise from
comparing the dark KdV system (27) with the integrable supersymmetric KdV
equation.
Type 5. Dual strange dark KdV system [6] related to the dual supersymmetric
KdV equation. The fifth type of dark KdV systems can be written as

$\left\\{\begin{array}[]{l}u_{t}=J_{1x},\\\
v_{t}=\tilde{Y}v+3auu_{x},\qquad\tilde{Y}\equiv(3u-\partial_{x}^{2})\partial_{x}.\end{array}\right.$
(29)

The dark KdV system (29) is integrable because of the existence of higher
order symmetries. The fifth order symmetry flow of (29) has the form,

$\left\\{\begin{array}[]{l}u_{\tau}=J_{2x},\\\
v_{\tau}=(5uv_{xx}-v_{xxxx})_{x}+5(u_{xx}-2u^{2})v_{x}+5a\big{(}uu_{xxx}+2u_{x}u_{xx}-4u^{2}u_{x}\big{)}.\end{array}\right.$
(30)

The operator $\tilde{Y}$ given in (29) is the dual operator of $-\hat{Y}$
defined in (27). In Secs. 5 and 9, we can find that the dark KdV system (29)
is also related to an integrable supersymmetric KdV equation.
Type 6. Mysterious dark KdV systems. The sixth type of dark KdV systems

$\left\\{\begin{array}[]{l}u_{t}=J_{x},\\\
v_{t}=-2\hat{Y}v+3aJ_{1x},\qquad\hat{Y}\equiv\partial_{x}(3u-\partial_{x}^{2}),\end{array}\right.$
(31)

with $a=0$ is called the mysterious dark KdV system [6]. The operator
$\hat{Y}$ in (31) is same as that of (27).

The fifth order symmetry flow of (31) reads

$\left\\{\begin{array}[]{l}u_{\tau}=J_{2x},\\\
v_{\tau}=-2[5vu_{xx}+10(uv_{x})_{x}-5u^{2}v-2v_{xxxx}]_{x}\\\
\qquad\quad+5a(8uu_{xxx}+18u_{x}u_{xx}-12u^{2}u_{x}-u_{xxxxx}).\end{array}\right.$
(32)

Type 7. Dual mysterious dark KdV systems. The seventh type of dark KdV systems
reads

$\left\\{\begin{array}[]{l}u_{t}=J_{1x},\\\
v_{t}=-2\tilde{Y}v+3aJ_{1}+3b(2u^{2}-u_{xx})_{x},\qquad\tilde{Y}\equiv(3u-\partial_{x}^{2})\partial_{x},\end{array}\right.$
(33)

which is a dual form of the mysterious dark KdV system (31) for $a=b=0$. The
operator $\tilde{Y}$ in (33) is same as that of (29). The dual mysterious dark
KdV system (33) is symmetry integrable with the following fifth order symmetry
flow

$\left\\{\begin{array}[]{lll}u_{\tau}&=&J_{2x},\\\
v_{\tau}&=&4v_{xxxxx}-20(uv_{xx})_{x}-10(u_{xx}-u^{2})v_{x}+5a(5u_{x}^{2}+8uu_{xx}\\\
&&-4u^{3}-u_{xxxx})+5b(6uu_{xx}+2u_{x}^{2}-u_{xxxx})_{x}-40bu^{2}u_{x}.\end{array}\right.$
(34)

Type 8. An extension of the completely decoupled dark KdV system. The eighth
type of dark KdV systems can be written as

$\left\\{\begin{array}[]{l}u_{t}=J_{1x},\\\
v_{t}=-\gamma_{1}v_{xxx}+\gamma_{1}(au+bu_{x})_{xx}+aJ_{1}+bJ_{1x},\end{array}\right.$
(35)

with the fifth order symmetry flow

$\left\\{\begin{array}[]{l}u_{\tau}=J_{2x},\\\
v_{\tau}=-\gamma_{2}v_{xxxxx}+\gamma_{1}(au+bu_{x})_{xxxx}+aJ_{2}+bJ_{2x},\end{array}\right.$
(36)

where $a,\ b$ and $\gamma_{1}$ are arbitrary constants while
$\gamma_{2}=\gamma_{1}$ for $a^{2}+b^{2}\neq 0$ and $\gamma_{2}$ is an
arbitrary constant for $a=b=0$.

According to the findings reported in Ref. [6], it has been established that
the dark KdV system represented by equation (35) and its corresponding fifth
order flow given by equation (36) exhibit complete decoupling between the
fields $u$ and $v$ when the parameters $a$ and $b$ are both set to zero. This
particular case aligns with the reference mentioned.
Type 9. First order dark KdV systems. The ninth type of dark KdV systems
includes three first order coupled nonhomogeneous linear systems

$\left\\{\begin{array}[]{l}u_{t}=J_{1x},\\\
v_{t}=a_{i}u_{x}v+2uv_{x}-b_{i}u_{xxx}-c_{i}uu_{x}-d_{i}u_{xx}-e_{i}u^{2},\
i=1,\ 2,\ 3,\end{array}\right.$ (37)

where the constants $\\{a_{i},\ b_{i},\ c_{i}\\},\ i=1,\ 2,\ 3$ can be taken
as

$\displaystyle\\{a_{1},\ b_{1},\ c_{1},\ d_{1},\ e_{1}\\}=\\{a,\ b,\ b(a-4),\
0,\ 0\\},$ (38) $\displaystyle\\{a_{2},\ b_{2},\ c_{2},\ d_{2},\
e_{2}\\}=\\{2,\ a,\ -2a,\ b,\ 0\\},$ (39) $\displaystyle\\{a_{3},\ b_{3},\
c_{3},\ d_{3},\ e_{3}\\}=\\{0,\ a,\ -4a,\ b,\ -b\\},$ (40)

with arbitrary constants $a$ and $b$, respectively.

The fifth order symmetry flow of (37) possesses the form

$\left\\{\begin{array}[]{l}u_{\tau}=J_{2x},\\\
v_{\tau}=\alpha_{i}J_{1x}v-2J_{1}v_{x}-\beta_{i}u_{xxxx}-\gamma_{i}uu_{xx}-\delta_{i}u_{x}^{2}-\theta_{i}u^{3}\\\
\qquad\quad-\mu_{i}u_{xxxxx}-\nu_{i}uu_{xxx}-\kappa_{i}u_{x}u_{xx}-\rho_{i}u^{2}u_{x},\end{array}\right.$
(41)

where the constants $\\{\alpha_{i},\ \beta_{i},\ \gamma_{i},\ \delta_{i},\
\theta_{i},\ \mu_{i},\ \nu_{i},\ \kappa_{i},\ \rho_{i}\\}$ corresponding to
(38)–(40) should be fixed as

$\displaystyle\\{\alpha_{1},\ \beta_{1},\ \gamma_{1},\ \delta_{1},\
\theta_{1},\ \mu_{1},\ \nu_{1},\ \kappa_{1},\ \rho_{1}\\}=\\{a,\ 0,\ 0,\ 0,\
0,\ b,\ b(a-10),\ -18b,\ 6b(4-a)\\},$ $\displaystyle\\{\alpha_{2},\
\beta_{2},\ \gamma_{2},\ \delta_{2},\ \theta_{2},\ \mu_{2},\ \nu_{2},\
\kappa_{2},\ \rho_{2}\\}=\\{-2,\ b,\ -6b,\ -6b,\ 0,\ a,\ -8a,\ -18a,\
-12a\\},$ $\displaystyle\\{\alpha_{3},\ \beta_{3},\ \gamma_{3},\ \delta_{3},\
\theta_{3},\ \mu_{3},\ \nu_{3},\ \kappa_{3},\ \rho_{3}\\}=\\{0,\ b,\ -8b,\
-5b,\ 4b,\ a,\ -10a,\ -18a,\ 24a\\}$

with $a$ and $b$ being arbitrary constants, respectively.

##  4 Dark KdV systems from Dark parameter expansions

Dark integrable systems can be derived through a variety of methodologies from
established integrable systems. In this section, we put forth a methodology
for obtaining a particular category of graded linear integrable systems by
introducing multiple dark parameters.

Assuming a dependence of the solutions to the KdV equation (12) on a single
dark parameter $\theta$ (or anyon parameters [16, 17]) that satisfies the
condition $\theta^{n+1}=0$, a dark KdV system beyond (4) for $n\geq 2$

$u_{it}=\left(3\sum_{j=0}^{i}u_{j}u_{i-j}-u_{ixx}\right)_{x},\ i=0,\ 1,\ 2,\
\ldots,\ n,$ (42)

by directly substituting can be derived by directly substituting the
corresponding expression for the dark parameter $\theta$

$u=\sum_{i=0}^{n}u_{i}\theta^{i}$ (43)

into the KdV equation (12).

This approach allows for the extension of the existing KdV system to
incorporate the influence of the dark parameter. The concept of a dark
parameter has been previously introduced in the literature [15], and similar
considerations have been made regarding anyon parameters [16, 17, 18].

The dark KdV system (42) corresponds to the integrable KdV coupling derived by
Ma and Fuchssteiner using a perturbation method [7]. The expansion (43) can
also be interpreted as representing $u$ as a vector in a specialized anyon
space, where the basis vectors are given by $\\{\theta^{i},\ i=0,\ 1,\
\ldots,\ n\\}$. It is worth noting that for the case of $n=1$, the dark
parameter coincides with the conventional Grassmann number.

The nonlinear dark KdV system (42) can be rewritten as

$\left\\{\begin{array}[]{l}u_{0t}=(3u_{0}^{2}-u_{0xx})_{x},\\\
\displaystyle{u_{it}=\partial_{x}\big{[}(6u_{0}-\partial_{x}^{2})\big{]}u_{i}+3\sum_{j=1}^{i-1}\left(u_{j}u_{i-j}\right)_{x},\
i=1,\ 2,\ \ldots,\ n,}\end{array}\right.$ (44)

which manifests itself (and then (42)) as a graded linear dark KdV system.

The graded linear dark KdV system (42) exhibits symmetry integrability,
indicating that it possesses a set of higher order symmetry flows. These
additional symmetries can be derived by employing the same expansion technique
to the higher order KdV equations. Specifically, the fifth symmetry flow of
the dark KdV system (42) can be expressed in form

$u_{i\tau}=\left[\sum_{j=0}^{i}\left(10u_{jxx}u_{i-j}+5u_{jx}u_{(i-j)x}\right)-10\sum_{j=0}^{i}\sum_{k=0}^{j}u_{k}u_{j-k}u_{i-j}-u_{ixxxx}\right]_{x},\
i=0,\ 1,\ \ldots,\ n.$ (45)

In the specific case where $\\{n=1,\ u_{0}=u,\ u_{1}=v\\}$, the dark KdV
system (42) and the fifth order KdV system (45) correspond to the type-two
case (22) and (23) with $a=b=0$.

For $n=2$, the dark KdV system (42) and the fifth order dark KdV system (45)
can be written as ($\\{u_{0},\ u_{1},\ u_{2}\\}\rightarrow\\{u,\ u,\ w\\}$)

$\left\\{\begin{array}[]{l}u_{t}=J_{1x},\\\ v_{t}=(6uv-v_{xx})_{x},\\\
w_{t}=(6uw+2v^{2}-w_{xx})_{x},\end{array}\right.$ (46)

and

$\left\\{\begin{array}[]{l}u_{t}=J_{2x},\\\
v_{t}=(10uv_{xx}+10vu_{xx}+10u_{x}v_{x}-30u^{2}v-v_{xxxx})_{x},\\\
w_{t}=[10uw_{xx}+10(wu_{x})_{x}+10vv_{xx}+5v_{x}^{2}-30u^{2}w-30uv^{2}-w_{xxxx}]_{x},\end{array}\right.$
(47)

respectively.

If additional dark parameters are incorporated into the solutions of the KdV
equation (12), it is possible to identify further integrable dark KdV systems.
For instance, when the solutions of the KdV equation (12) rely on two dark
parameters, namely $\theta_{1}$ and $\theta_{2}$, satisfying the conditions
$\theta_{1}^{n+1}=0$ and $\theta_{2}^{m+1}=0$, a dark KdV system emerges

$u_{ikt}=\left(3\sum_{j=0}^{i}\sum_{l=0}^{m}u_{jl}u_{(i-j)(k-l)}-u_{ikxx}\right)_{x},\
i=0,\ 1,\ \ldots,\ n,\ k=0,\ 1,\ \ldots,m,$ (48)

which is related to the KdV equation by

$u=\sum_{i=0}^{n}\sum_{k=0}^{m}u_{ik}\theta_{1}^{i}\theta_{2}^{k}.$ (49)

##  5 Dark KdV systems from the bosonizations of integrable supersymmetric
fermion systems

The introduction section highlights the significance of supersymmetry in
addressing various important physical phenomena, including dark matter [9, 10,
11]. In references [12, 20], a bosonization approach is proposed as a means to
solve supersymmetric systems. By employing the bosonization method, it becomes
possible to solve supersymmetric integrable systems using certain types of
dark boson systems. For example, in the case of the supersymmetric KdV
equation

$\Phi_{t}=\big{[}3({\cal D}\Phi)\Phi-\Phi_{xx}\big{]}_{x},\ \Phi=\Phi(x,\ t,\
\theta)=\xi(x,t)+w(x,\ t)\theta,$ (50)

where $\Phi\equiv\Phi(x,\ t,\ \theta)$ is a fermionic superfield, $\xi(x,t)$
is an anticommuting field and ${\cal{D}}=\partial_{\theta}+\theta\partial_{x}$
is the supersymmetric derivative, an abundance of dark KdV systems can be
identified [20].

The dark KdV equation (22) with $a=b=0$ represents the simplest instance with
the relation $\Phi=\zeta v+u\theta,\ \zeta^{2}=0$. The next three dark KdV
systems read

$\left\\{\begin{array}[]{l}u_{t}=J_{1x},\\\ v_{i,t}=\hat{Y}v_{i},\ i=1,\ 2,\\\
v_{12t}=\hat{K}v_{12}+3(v_{2}v_{1x}-v_{1}v_{2x})_{x},\end{array}\right.$ (51)
$\left\\{\begin{array}[]{l}u_{t}=J_{1x},\\\ v_{it}=\hat{Y}v_{i},\ i=1,\ 2,\
3\\\ v_{ijt}=\hat{K}v_{ij}+3(v_{j}v_{ix}-v_{i}v_{jx})_{x},\ i,j=1,\ 2,\
3,j>i,\\\
v_{123t}=\hat{Y}v_{123}+3(v_{1}v_{23}-v_{2}v_{13}+v_{3}{v_{12}})_{x},\end{array}\right.$
(52)

and

$\left\\{\begin{array}[]{l}u_{t}=J_{1x},\\\ v_{it}=\hat{Y}v_{i},\ i=1,\ 2,\
3,\ 4,\\\ v_{ijt}=\hat{K}v_{ij}+3(v_{ix}v_{j}-v_{i}v_{jx})_{x},\ i,\ j=1,\ 2,\
3,\ 4,\ j>i,\\\
v_{ijkt}=\hat{Y}v_{ijk}+3(v_{i}v_{jk}-v_{j}v_{ik}+v_{ij}v_{k})_{x},\ i,\ j,\
k=1,\ 2,\ 3,\ 4,\ i<j<k,\\\
v_{1234t}=\hat{K}v_{1234}+6(v_{12}v_{34}+v_{14}v_{23}-v_{13}v_{24})_{x}+3(v_{123}v_{4x}\\\
\qquad\qquad-
v_{123x}v_{4}+v_{124}v_{3x}-v_{124x}v_{3}+v_{134}v_{2x}-v_{134x}v_{2}+v_{234}v_{1x}-v_{234x}v_{1})_{x},\end{array}\right.$
(53)

where the operators $\hat{Y}$ and $\hat{K}$ are defined in (27) and (19),
respectively.

The exact solutions of the supersymmetric KdV equation (50) are related to the
dark KdV systems by $\Phi=\xi+w\theta$ with

$\xi=v_{1}\zeta_{1}+v_{2}\zeta_{2},\ w=u+v_{12}\zeta_{1}\zeta_{2}$ (54)

for (51),

$\xi=v_{1}\zeta_{1}+v_{2}\zeta_{2}+v_{3}\zeta_{3}+v_{123}\zeta_{1}\zeta_{2}\zeta_{3},\
w=u+v_{12}\zeta_{1}\zeta_{2}+v_{13}\zeta_{1}\zeta_{3}+v_{23}\zeta_{2}\zeta_{3}$
(55)

for (52) and

$\displaystyle
w=u+v_{12}\zeta_{1}\zeta_{2}+v_{13}\zeta_{1}\zeta_{3}+v_{14}\zeta_{1}\zeta_{4}+v_{23}\zeta_{2}\zeta_{3}+v_{24}\zeta_{2}\zeta_{4}+v_{34}\zeta_{3}\zeta_{4}+v_{1234}\zeta_{1}\zeta_{2}\zeta_{3}\zeta_{4},$
$\displaystyle\xi=v_{1}\zeta_{1}+v_{2}\zeta_{2}+v_{3}\zeta_{3}+v_{4}\zeta_{4}+v_{123}\zeta_{1}\zeta_{2}\zeta_{3}+v_{124}\zeta_{1}\zeta_{2}\zeta_{4}+v_{134}\zeta_{1}\zeta_{3}\zeta_{4}+v_{234}\zeta_{2}\zeta_{3}\zeta_{4}$
(56)

for (53), respectively, where $\zeta_{i},\ i=1,\ 2,\ 3,\ 4$ are four different
Grassmann parameters and $u,\ v_{i},\ v_{ij},\ v_{ijk},i<j<k<l,\ i,\ j,\ k=1,\
2,\ 3,\ 4$ and $v_{1234}$ are sixteen dark boson fields.

Another simple integrable supersymmetric fermionic KdV equation possesses the
form

$\Phi_{t}=-\Phi_{xxx}+6({\cal D}\Phi)\Phi_{x},\ \Phi=\xi+w\theta.$ (57)

Utilizing the bosonization method (5) on the supersymmetric KdV equation (57),
it is possible to derive a series of integrable dark KdV systems

$\displaystyle\left\\{\begin{array}[]{l}u_{t}=J_{1x},\\\
v_{it}=\tilde{K}v_{i},\ \tilde{K}\equiv(6u-\partial_{x}^{2})\partial_{x},\\\
v_{ijt}=\hat{K}v_{ij},\ i,\ j=1,\ 2,\ 3,\ 4,\ i<j,\\\
v_{ijkt}=\tilde{K}v_{ijk}+6(v_{ix}v_{jk}-v_{jx}v_{ik}+v_{ij}v_{kx}),\ i<j<k,\
i,\ j,\ k=1,\ 2,\ 3,\ 4,\\\
v_{1234t}=\hat{K}v_{1234}+6(v_{12}v_{34}-v_{13}v_{24}+v_{14}v_{23})_{x},\end{array}\right.$
(63)

where the operators $\hat{K}$ and $\tilde{K}$ are same as those defined in
(19) and (24). The simplest trivial case of (63) with
$v_{2}=v_{3}=v_{4}=v_{ij}=v_{ijk}=v_{1234}=0$ and $v_{1}\rightarrow v$ is just
the dark KdV system (25) for $a=b=0$.

The triviality in these examples comes from the fact that the resulting
systems are linear in the fermionic field and, as a result, the first equation
of these systems is simply the KdV equation itself.

The integrability of the dark systems (51), (52), (53), and (63) is evident as
they emerge as specific reductions of the integrable supersymmetric KdV
systems (50) and (57). Notably, the higher order symmetries of (51), (52),
(53), and (63) can be derived by employing the same bosonization procedure
applied to the higher order supersymmetry KdV equations [19] associated with
the supersymmetric KdV equations (50) and (57).

##  6 Dark KdV systems from supersymmetric integrable boson systems

By exploring different variants of supersymmetric KdV equations and coupled
KdV systems, one can uncover a broader range of integrable dark systems.

For instance, the supersymmetric bosonic KdV equation,

$U_{t}=(3U^{2}-U_{xx})_{x},\ U(x,\ t,\ \theta)=w(x,\ t)+\xi(x,\ t)\theta,$
(64)

also allows infinitely many dark KdV systems. One of the simple integrable
dark KdV system reads

$\displaystyle\left\\{\begin{array}[]{l}u_{t}=J_{1x},\\\ v_{it}=\hat{K}v_{i},\
i=1,\ 2,\ 3,\ 4,\\\ v_{ijt}=\hat{K}v_{ij},\ i,\ j=1,\ 2,\ 3,\ 4,\ i<j,\\\
v_{ijkt}=\hat{K}v_{ijk}+6(v_{i}v_{jk}-v_{j}v_{ik}+v_{ij}v_{k})_{x},\ i<j<k,\
i,\ j,\ k=1,\ 2,\ 3,\ 4,\\\
v_{1234t}=\hat{K}v_{1234}+6(v_{12}v_{34}-v_{13}v_{24}+v_{14}v_{23})_{x}\end{array}\right.$
(70)

with sixteen dark boson fields $\\{u,\ v_{i},\ v_{ij},\ v_{ijk},\ v_{ijkl},\
i,j,k,l=1,\ \ldots,\ 4,\ i<j<k<l\\}$.

The dark KdV system (70) is related to the supersymmetric KdV equation (64) by
$U=w+\xi\theta$ with the same $w$ and $\xi$ given in (5).

The generalized supersymmetric boson KdV system possesses the form (in
potential form)

$U_{t}=3U_{x}^{2}+c({\cal{D}}U_{x})({\cal{D}}U)-U_{xxx},\ U(x,\ t,\
\theta)=p(x,\ t)+\xi(x,\ t)\theta.$ (71)

In terms of the component fields, (71) is equivalent to

$\left\\{\begin{array}[]{l}w_{t}=(3w^{2}-w_{xx})_{x}+c\xi_{xx}\xi,\quad
w=p_{x},\\\ \xi_{t}=(cw\xi-\xi_{xx})_{x}+2(3-c)w\xi_{x}.\end{array}\right.$
(72)

Applying the bosonization formulae (5) to (72), we have

$\displaystyle\left\\{\begin{array}[]{l}u_{t}=J_{1x},\\\
v_{it}=\hat{K}_{1}v_{i},\
\hat{K}_{1}=cu_{x}+(6-c)u\partial_{x}-\partial_{x}^{3},\\\
v_{ijt}=\hat{K}v_{ij}+c(v_{ix}v_{j}-v_{i}v_{jx})_{x},\ i,\ j,\ k=1,\ 2,\ 3,\
4,\ i<j<k,\\\
v_{ijkt}=\hat{K}_{1}v_{ijk}+(6-c)(v_{ix}v_{jk}-v_{jx}v_{ik}+v_{ij}v_{kx})+c(v_{i}v_{jkx}-v_{j}v_{ikx}+v_{ijx}v_{k}),\\\
v_{1234t}=\hat{K}v_{1234}+6(v_{12}v_{34}-v_{13}v_{24}+v_{14}v_{23})_{x}+c(v_{123x}v_{4}\\\
\qquad\qquad-
v_{123}v_{4x}-v_{124x}v_{3}+v_{124}v_{3x}+v_{134x}v_{2}-v_{134}v_{2x}-v_{234x}v_{1}+v_{234}v_{1x})_{x}.\end{array}\right.$
(79)

The special cases of $c=3$ and $c=0$ hold particular significance in the
context under consideration. When $c=3$, it is noteworthy that
$\hat{K}_{1}=\hat{Y}$, and equation (79) is found to be identical to equation
(53). This observation leads to the implication that the supersymmetric boson
field equation (71) with $c=3$ is indeed equivalent to the supersymmetric
fermion field equation (50). Similarly, for the case of $c=0$, it is
highlighted that $\hat{K}_{1}=\tilde{K}$, resulting in the reduction of
equation (79) back to equation (63). These findings reveal significant
relationships and equivalences between the equations for different values of
$c$. The implications of these relationships are contingent upon the specific
mathematical and physical context in which these equations are being examined.

##  7 Dark KdV systems from the bosonization of the superintegrable KdV system

The extension of the bosonization technique broadens its applicability and
offers a valuable framework for studying diverse fermionic systems, including
the super-integrable KdV system [13, 21, 22]

$\left\\{\begin{array}[]{l}w_{t}=(3w^{2}-w_{xx}+3\xi\xi_{x})_{x},\\\
\xi_{t}=-4\xi_{xxx}+6w\xi_{x}+3w_{x}\xi,\end{array}\right.$ (80)

where $\xi$ is a fermion field and $w$ is a boson field.

Applying the same bosonization solution (5) to the super-integrable KdV system
(80), we get a new type of integrable dark KdV system,

$\displaystyle\left\\{\begin{array}[]{l}u_{t}=J_{1x},\\\ v_{it}=\hat{L}v_{i},\
\hat{L}\equiv-4\partial_{x}^{3}+6u\partial_{x}+3u_{x},\ i=1,\ 2,\ 3,\ 4,\\\
v_{ijt}=\hat{K}v_{ij}+3(v_{i}v_{jx}-v_{ix}v_{j})_{x},\ i,\ j,\ k=1,\ 2,\ 3,\
4,\ i<j<k,\\\
v_{ijk}=\hat{L}v_{ijk}+6(v_{jk}v_{ix}-v_{ik}v_{jx}+v_{ij}v_{kx})+3(v_{jkx}v_{x}-v_{ikx}v_{j}+v_{ijx}v_{k}),\\\
v_{1234t}=\hat{K}v_{1234}+6(v_{12}v_{34}-v_{13}v_{24}+v_{14}v_{23})_{x}+3(v_{1}v_{234x}-v_{1x}v_{234}\\\
\qquad\qquad-
v_{2}v_{134x}+v_{2x}v_{134}+v_{3}v_{124x}-v_{3x}v_{124}-v_{4}v_{123x}+v_{4x}v_{123})_{x}.\end{array}\right.$
(87)

Taking $v_{2}=v_{3}=v_{4}=0,\ v_{ij}=v_{ijk}=0,i,\ j,\ k=1,\ 2,\ 3,\ 4$ and
$v_{1234}=0$, the dark KdV system (87) is reduced back to the two-component
dark system (17) with $a=0$.

A four-component integrable dark KdV system and an eight-component integrable
dark KdV system can be simply read off from (87) by taking
$v_{3}=v_{4}=v_{13}=v_{14}=v_{23}=v_{24}=v_{34}=v_{123}=v_{124}=v_{134}=v_{234}=v_{1234}=0$
and $v_{4}=v_{14}=v_{24}=v_{34}=v_{124}=v_{134}=v_{234}=v_{1234}=0$,
respectively.

##  8 Nonlinear dark KdV systems

In this section, we extend the linear dark systems (4) to nonlinear systems
(9) for the KdV equation. Moreover, when studying higher-dimensional
integrable systems such as the (2+1)-dimensional KP, BKP, and CKP equations,
it is feasible to identify diverse lower-dimensional integrable systems, some
of which may exhibit characteristics akin to (extended) dark equations. For
instance, to solve the BKP equation [14]

$v_{xt}+(v_{xxxx}+15vv_{xx}+15v^{3}-15vq-5v_{xy})_{xx}-5v_{yy},\ q_{x}=v_{y},$
(88)

one can find the following nonlinear dark KdV system

$\left\\{\begin{array}[]{l}u_{y}=\left(u_{xx}-\frac{a}{2}u^{2}\right)_{x},\\\
v_{y}=\left(v_{xx}+3v^{2}+auv+\frac{a^{2}}{6}\right)_{x}\end{array}\right.$
(89)

with the fifth order flow ($w=au+3v$),

$\left\\{\begin{array}[]{l}u_{t}=\left(9u_{xxxx}-15auu_{xx}-\frac{15a}{2}u_{x}^{2}+\frac{5a^{2}}{2}u^{3}\right)_{x},\\\
v_{t}=\left[9v_{xxxx}+15(w+3v)v_{xx}+5awu_{xx}+15w_{x}v_{x}+\frac{15v}{2}(w^{2}+3v^{2})+\frac{5a^{2}}{2}u_{x}^{2}\right]_{x}.\end{array}\right.$
(90)

A direct proof can establish that if $u$ and $v$ are solutions of the dark KdV
system (89), then $v$ satisfies the BKP equation (88). In Refs. [23, 14], it
has been established that when $a=-3$ or $a=-6$, both $u$ and $v$ serve as
solutions to the BKP equation (88).

##  9 Supersymmetric dark KdV systems

The previous sections (sections 5 and 6) have revealed that within a single
integrable supersymmetric model, the bosonization method allows for the
identification of diverse dark fields. These dark fields are characterized by
satisfying graded linear dark equations. In this section, we delve deeper into
the investigation of supersymmetric dark KdV systems.

The most general possible supersymmetric extension of the KdV equation is a
one-parameter family of equations with only one fermionic field and with no
inverse power of the fields

$\Phi_{t}=-\Phi_{xxx}+a({\cal{D}}\Phi_{x})\Phi+(6-a)({\cal{D}}\Phi)\Phi_{x}.$
(91)

A comprehensive Painlevé analysis indicates that within the class of one-
parameter systems, only two distinctive cases, specifically equation (50) for
$a=3$ and equation (57) for $a=0$, exhibit integrability. Notably, an
additional significant characteristic of these two equations is their ability
to be formulated as Hamiltonian systems, employing the Poisson brackets
associated with the second Hamiltonian structure [24].

###  9.1 Supersymmetric dark KdV systems related to the supersymmetric KdV
equation (50)

To construct the most comprehensive supersymmetric dark KdV system, in a
manner akin to the classical integrable cases, it is necessary to consider a
linear combination of all possible terms which ensures that all relevant terms
and interactions are taken into account

$\left\\{\begin{array}[]{l}\Phi_{t}=-\Phi_{xxx}+3[({\cal{D}}\Phi)\Phi]_{x},\\\
\Psi_{t}=-a_{1}\Psi_{xxx}+a_{2}\Phi{\cal{D}}\Psi_{x}+a_{3}({\cal{D}}\Phi_{x})\Psi+a_{4}\Phi_{x}{\cal{D}}\Psi+a_{5}({\cal{D}}\Phi)\Psi_{x}\end{array}\right.$
(92)

such that it is symmetry integrable.

Upon completing arduous computations, one can ascertain the existence of
supersymmetric integrable dark KdV systems characterized by the presence of
higher order symmetries.

Case 1. The first type of supersymmetric dark KdV systems is characterized by
a specific form

$\left\\{\begin{array}[]{l}\Phi_{t}=-\Phi_{xxx}+3(\Phi{\cal{D}}\Phi)_{x},\\\
\Psi_{t}=-\Psi_{xxx}+3\Phi{\cal{D}}\Psi_{x},\end{array}\right.$ (93)

where $\Psi$ is also a fermionic super field. The symmetry integrability of
(93) is guaranteed by the existence of higher order symmetry

$\left\\{\begin{array}[]{l}\Phi_{\tau}=-\Phi_{xxxxx}+\big{[}5(\Phi_{x}{\cal{D}}\Phi)_{x}+5\Phi{\cal{D}}\Phi_{xx}-10\Phi({\cal{D}}\Phi)^{2}\big{]}_{x},\\\
\Psi_{\tau}=-\Psi_{xxxxx}+5(\Phi{\cal{D}}\Psi_{xx})_{x}+5(\Phi_{xx}-2\Phi{\cal{D}}\Phi){\cal{D}}\Psi_{x}.\end{array}\right.$
(94)

In the supersymmetric case, there are two types of conservation laws. One
possesses the same forms

$\rho_{t}=J_{x},$ (95)

as those for the usual integrable systems, and the other can be written as

$\rho_{t}={\cal{DF}},$ (96)

where $\rho$ is the conserved density while $J$ and ${\cal{F}}$ are the
conserved flows related to the first and second types of conservation laws,
respectively. It is clear that $\rho=\Phi$ and
$J=-\Phi_{xx}+3\Phi{\cal{D}}\Phi$ is related to the first type of the
conservation laws (95). From (93), one can also check that

$\rho=\Psi_{x}\Phi,\
{\cal{F}}={\cal{D}}(\Psi_{xx}\Phi_{x}-\Psi_{x}\Phi_{xx}-\Psi_{xxx}\Phi)+3\Phi\Phi_{x}\Psi_{x}$
(97)

are related to the second type of conservation laws (96).
Case 2. The second supersymmetric dark KdV equation reads

$\left\\{\begin{array}[]{l}\Phi_{t}=-\Phi_{xxx}+3(\Phi{\cal{D}}\Phi)_{x},\\\
\Psi_{t}=-\Psi_{xxx}+3\Psi_{x}{\cal{D}}\Phi,\end{array}\right.$ (98)

with the fifth order supersymmetric flow

$\left\\{\begin{array}[]{l}\Phi_{\tau}=-\Phi_{xxxxx}+\big{[}5(\Phi_{x}{\cal{D}}\Phi)_{x}+5\Phi{\cal{D}}\Phi_{xx}-10\Phi({\cal{D}}\Phi)^{2}\big{]}_{x},\\\
\Psi_{\tau}=-\Psi_{xxxxx}+5(\Psi_{xx}{\cal{D}}\Phi)_{x}+5[{\cal{D}}\Phi_{xx}-2({\cal{D}}\Phi)^{2}+\Phi\Phi_{x}]\Psi_{x}.\end{array}\right.$
(99)

Case 3. The third supersymmetric dark KdV system can be expressed as

$\left\\{\begin{array}[]{l}\Phi_{t}=-\Phi_{xxx}+3(\Phi{\cal{D}}\Phi)_{x},\\\
\Psi_{t}=-\Psi_{xxx}+3\Phi{\cal{D}}\Psi_{x}+3\Psi_{x}{\cal{D}}\Phi.\end{array}\right.$
(100)

In this case, one can prove that $\rho=\Psi\Phi$ is a conserved density with
the first type of conservation flow (95),

$(\Psi\Phi)_{t}=(-\Psi_{xx}\Phi-\Psi\Phi_{xx}+\Psi_{x}\Phi_{x}+\Psi\Phi{\cal{D}}\Phi)_{x}.$

The integrability of (100) is guaranteed by the existence of higher order
symmetries, for instance, the fifth order symmetry possesses the form

$\left\\{\begin{array}[]{l}\Phi_{\tau}=-\Phi_{xxxxx}+5\big{[}(\Phi_{x}{\cal{D}}\Phi)_{x}+\Phi{\cal{D}}\Phi_{xx}-2\Phi({\cal{D}}\Phi)^{2}\big{]}_{x},\\\
\Psi_{\tau}=-\Psi_{xxxxx}+5(\Phi{\cal{D}}\Psi_{xx}+\Psi_{xx}{\cal{D}}\Phi)_{x}+5[{\cal{D}}\Phi_{xx}-2({\cal{D}}\Phi)^{2}]\Psi_{x}+5(\Phi_{xx}-4\Phi{\cal{D}}\Phi){\cal{D}}\Psi_{x}.\end{array}\right.$
(101)

Case 4. The fourth type of supersymmetric dark KdV systems has the following
simple form

$\left\\{\begin{array}[]{l}\Phi_{t}=-\Phi_{xxx}+3(\Phi{\cal{D}}\Phi)_{x},\\\
\Psi_{t}=-\Psi_{xxx}+3{\cal{D}}(\Phi\Psi_{x})\end{array}\right.$ (102)

accompanied by the fifth order supersymmetric flow

$\left\\{\begin{array}[]{l}\Phi_{\tau}=-\Phi_{xxxxx}+5\big{[}(\Phi_{x}{\cal{D}}\Phi)_{x}+\Phi{\cal{D}}\Phi_{xx}-2\Phi({\cal{D}}\Phi)^{2}\big{]}_{x},\\\
\Psi_{\tau}=-\Psi_{xxxxx}+5{\cal{D}}[\Phi\Psi_{xxx}+(\Phi_{x}\Psi_{x})_{x}-2\Phi({\cal{D}}\Phi)\Psi_{x}].\end{array}\right.$
(103)

In this case the dark field $\Psi$ itself is a conserved density with the
second type of flow ${\cal{F}}=3\Phi\Psi_{x}-{\cal{D}}\Psi_{xx}$.
Case 5. The fifth supersymmetric dark KdV system

$\left\\{\begin{array}[]{l}\Phi_{t}=-\Phi_{xxx}+3(\Phi{\cal{D}}\Phi)_{x},\\\
\Psi_{t}=-\Psi_{xxx}+3(\Phi{\cal{D}}\Psi)_{x}\end{array}\right.$ (104)

possesses the fifth order symmetry,

$\left\\{\begin{array}[]{l}\Phi_{\tau}=-\Phi_{xxxxx}+5\big{[}(\Phi_{x}{\cal{D}}\Phi)_{x}+\Phi{\cal{D}}\Phi_{xx}-2\Phi({\cal{D}}\Phi)^{2}\big{]}_{x},\\\
\Psi_{\tau}=-\Psi_{xxxxx}+5[(\Phi{\cal{D}}\Psi_{x})_{x}+\Phi_{xx}{\cal{D}}\Psi-2\Phi({\cal{D}}\Psi)({\cal{D}}\Phi)]_{x}.\end{array}\right.$
(105)

In the previous four cases, (93), (98), (100), and (102), the dark field
$\Psi$ is not a first type of conserved density. (104) is the first case with
the first type of conserved flow $J=3\Phi{\cal{D}}\Psi-\Psi_{xx}$.
Case 6. The sixth type supersymmetric dark KdV system

$\left\\{\begin{array}[]{l}\Phi_{t}=-\Phi_{xxx}+3(\Phi{\cal{D}}\Phi)_{x},\\\
\Psi_{t}=-\Psi_{xxx}+3\big{(}\Psi{\cal{D}}\Phi\big{)}_{x}\end{array}\right.$
(106)

represents a conserved form for the dark field $\Psi$ and its associated fifth
order symmetry flow reads

$\left\\{\begin{array}[]{l}\Phi_{\tau}=-\Phi_{xxxxx}+\big{[}5\Phi_{xx}{\cal{D}}\Phi+5\Phi_{x}{\cal{D}}\Phi_{x}+5\Phi{\cal{D}}\Phi_{xx}-10\Phi({\cal{D}}\Phi)^{2}\big{]}_{x},\\\
\Psi_{\tau}=5\big{[}\Psi{\cal{D}}\Phi_{xx}+\Psi_{x}{\cal{D}}\Phi_{x}+\Psi_{xx}{\cal{D}}\Phi+\Phi\Phi_{x}\Psi-2({\cal{D}}\Phi)^{2}\Psi\big{]}_{x}-\Psi_{xxxxx}.\end{array}\right.$
(107)

In this case, the dark field $\Psi$ is also a first type of conserved density
with the conserved flow $J=-\Psi_{xx}+3\Psi{\cal{D}}\Phi$.
Case 7. The seventh supersymmetric dark KdV system can be written as

$\left\\{\begin{array}[]{l}\Phi_{t}=-\Phi_{xxx}+3(\Phi{\cal{D}}\Phi)_{x},\\\
\Psi_{t}=-\Psi_{xxx}+3{\cal{D}}[({\cal{D}}\Phi)({\cal{D}}\Psi)],\end{array}\right.$
(108)

along with the fifth order supersymmetric flow

$\left\\{\begin{array}[]{l}\Phi_{\tau}=-\Phi_{xxxxx}+5\big{[}(\Phi_{x}{\cal{D}}\Phi)_{x}+\Phi{\cal{D}}\Phi_{xx}-2\Phi({\cal{D}}\Phi)^{2}\big{]}_{x},\\\
\Psi_{\tau}=-\Psi_{xxxxx}+5{\cal{D}}\big{\\{}[({\cal{D}}\Phi){\cal{D}}\Psi_{x}]_{x}+[{\cal{D}}\Phi_{xx}+\Phi\Phi_{x}-2({\cal{D}}\Phi)^{2}]{\cal{D}}\Psi\big{\\}}.\end{array}\right.$
(109)

In this case, the dark field $\Psi$ is a conserved density with the second
type of conserved flow
${\cal{F}}=-{\cal{D}}\Psi_{xx}+3({\cal{D}}\Phi){\cal{D}}\Psi$.
Case 8. The eighth type of supersymmetric KdV systems

$\left\\{\begin{array}[]{l}\Phi_{t}=-\Phi_{xxx}+3(\Phi{\cal{D}}\Phi)_{x},\\\
\Psi_{t}=-\Psi_{xxx}+3\Psi_{x}{\cal{D}}\Phi+3{\cal{D}}(\Phi\Psi_{x}).\end{array}\right.$
(110)

possesses a second type of conservation laws

$(\Psi\Phi_{x})_{t}={\cal{D}}\left[{\cal{D}}\left(\Phi_{xxx}\Psi-\Psi_{xx}\Phi_{x}+\Psi_{x}\Phi_{xx}+3\Psi\Phi{\cal{D}}\Phi_{x}+3\Psi\Phi_{x}{\cal{D}}\Phi\right)-3\Phi\Phi_{x}\Psi_{x}\right].$

The fifth order symmetry flow of (109) reads

$\left\\{\begin{array}[]{l}\Phi_{\tau}=5\big{[}(\Phi_{xx}{\cal{D}}\Phi)_{x}+\Phi{\cal{D}}\Phi_{xx}-2\Phi({\cal{D}}\Phi)^{2}\big{]}_{x}-\Phi_{xxxxx},\\\
\Psi_{\tau}=5(2\Psi_{xx}{\cal{D}}\Phi-\Phi{\cal{D}}\Psi_{xx})_{x}-5(\Phi_{xx}-4\Phi{\cal{D}}\Phi){\cal{D}}\Psi_{x}\\\
\qquad\quad+10[{\cal{D}}\Phi_{xx}-3({\cal{D}}\Phi)^{2}+2\Phi\Phi_{x}]\Psi_{x}-\Psi_{xxxxx}.\end{array}\right.$
(111)

Case 9. For the ninth type of supersymmetric dark KdV equation systems

$\left\\{\begin{array}[]{l}\Phi_{t}=-\Phi_{xxx}+3(\Phi{\cal{D}}\Phi)_{x},\\\
\Psi_{t}=-\Psi_{xxx}+3(\Phi{\cal{D}}\Psi)_{x}+3{\cal{D}}[({\cal{D}}\Phi)({\cal{D}}\Psi)],\end{array}\right.$
(112)

the dark field $\Psi$ is conserved with the second type of conserved flow
${\cal{F}}={\cal{D}}(3\Phi{\cal{D}}\Psi-\Psi_{xx})+3({\cal{D}}\Phi){\cal{D}}\Psi$.
The fifth order symmetry flow of (112) possesses the form

$\left\\{\begin{array}[]{l}\Phi_{\tau}=-\Phi_{xxxxx}+5\big{[}(\Phi_{x}{\cal{D}}\Phi)_{x}+\Phi{\cal{D}}\Phi_{xx}-2\Phi({\cal{D}}\Phi)^{2}\big{]}_{x},\\\
\Psi_{\tau}=-\Psi_{xxxxx}+5[\Phi{\cal{D}}\Psi_{xx}+\Psi_{xx}{\cal{D}}\Phi+\Phi_{x}{\cal{D}}\Psi_{x}+2\Phi_{xx}{\cal{D}}\Psi-4({\cal{D}}\Psi)({\cal{D}}\Phi)\Phi]_{x}\\\
\qquad\quad+5{\cal{D}}[\Phi_{xx}\Psi_{x}-\Phi_{x}\Psi_{xx}+({\cal{D}}\Phi_{x})({\cal{D}}\Psi_{x})-2({\cal{D}}\Phi)^{2}{\cal{D}}\Psi].\end{array}\right.$
(113)

Case 10. The tenth type of supersymmetric dark KdV systems

$\left\\{\begin{array}[]{l}\Phi_{t}=-\Phi_{xxx}+3(\Phi{\cal{D}}\Phi)_{x},\\\
\Psi_{t}=-\Psi_{xxx}+3{\cal{D}}[\Phi\Psi_{x}+({\cal{D}}\Phi)({\cal{D}}\Psi)]\end{array}\right.$
(114)

is symmetry integrable because of the existence of the higher order
symmetries, say,

$\left\\{\begin{array}[]{l}\Phi_{\tau}=-\Phi_{xxxxx}+5\big{[}(\Phi_{x}{\cal{D}}\Phi)_{x}+\Phi{\cal{D}}\Phi_{xx}-2\Phi({\cal{D}}\Phi)^{2}\big{]}_{x},\\\
\Psi_{\tau}=-\Psi_{xxxxx}+5(\Psi_{xx}{\cal{D}}\Phi+\Psi_{x}{\cal{D}}\Phi_{x})_{x}-5{\cal{D}}[\Phi\Psi_{xxx}\\\
\qquad\quad-({\cal{D}}\Phi_{xx}){\cal{D}}\Psi+4\Phi({\cal{D}}\Phi)\Psi_{x}+2({\cal{D}}\Phi)^{2}{\cal{D}}\Psi].\end{array}\right.$
(115)

In this case, the dark field $\Psi$ is also conserved with the second type of
conserved flow

${\cal{F}}=-{\cal{D}}\Psi_{xx}+3\Phi\Psi_{x}+3({\cal{D}}\Phi){\cal{D}}\Psi.$

Case 11. The validity of the eleventh type of supersymmetric dark systems

$\left\\{\begin{array}[]{l}\Phi_{t}=-\Phi_{xxx}+3(\Phi{\cal{D}}\Phi)_{x},\\\
\Psi_{t}=-\Psi_{xxx}+3(\Phi{\cal{D}}\Psi+\Psi{\cal{D}}\Phi)_{x}\end{array}\right.$
(116)

is naturally established due to the direct correspondence between the dark
field $\Psi$ equation (the second equation of (116)) and the symmetry equation
of the $\Phi$ equation (the first equation of (116)). This correspondence
indicates that the dark field $\Psi$ serves as a conserved density,
accompanied by the first type of conserved flow
$J=-\Psi_{xx}+3\Phi{\cal{D}}\Psi+3\Psi{\cal{D}}\Phi$.

The higher order equations for the dark field $\Psi$ are also the symmetry
equations of the higher order supersymmetric KdV equations, say, the fifth
order symmetry flow of (116) reads

$\left\\{\begin{array}[]{l}\Phi_{\tau}=-\Phi_{xxxxx}+5\big{[}(\Phi_{x}{\cal{D}}\Phi)_{x}+\Phi{\cal{D}}\Phi_{xx}-2\Phi({\cal{D}}\Phi)^{2}\big{]}_{x},\\\
\Psi_{\tau}=5[(\Psi{\cal{D}}\Phi_{x}+\Phi{\cal{D}}\Psi_{x})_{x}+\Psi_{xx}{\cal{D}}\Phi+\Phi_{xx}{\cal{D}}\Psi-4\Phi({\cal{D}}\Phi){\cal{D}}\Psi-2({\cal{D}}\Phi)^{2}\Psi]_{x}-\Psi_{xxxxx}.\end{array}\right.$
(117)

Case 12. The twelfth type of supersymmetric dark KdV systems

$\left\\{\begin{array}[]{l}\Phi_{t}=-\Phi_{xxx}+3(\Phi{\cal{D}}\Phi)_{x},\\\
\Psi_{t}=-\Psi_{xxx}+3(\Psi{\cal{D}}\Phi-\Phi{\cal{D}}\Psi)_{x},\end{array}\right.$
(118)

closely resembles the eleventh type (116), except for a change in the sign of
the last term. The dark field $\Psi$ is also a conserved density with the
first type of conserved flow
$J=-\Psi_{xx}+3\Psi{\cal{D}}\Phi-3\Phi{\cal{D}}\Psi$.

The fifth order symmetry flow of (118) can be written as

$\left\\{\begin{array}[]{l}\Phi_{\tau}=-\Phi_{xxxxx}+5\big{[}(\Phi_{x}{\cal{D}}\Phi)_{x}+\Phi{\cal{D}}\Phi_{xx}-2\Phi({\cal{D}}\Phi)^{2}\big{]}_{x},\\\
\Psi_{\tau}=5\big{[}(\Psi{\cal{D}}\Phi_{x}-\Phi_{x}{\cal{D}}\Psi)_{x}+\Psi_{xx}{\cal{D}}\Phi-\Phi{\cal{D}}\Psi_{xx}+2\Phi\Phi_{x}\Psi\\\
\qquad\quad+2\Phi({\cal{D}}\Phi){\cal{D}}\Psi-2({\cal{D}}\Phi)^{2}\Psi\big{]}_{x}-\Psi_{xxxxx}.\end{array}\right.$
(119)

Case 13. The thirteenth supersymmetric dark KdV system,

$\left\\{\begin{array}[]{l}\Phi_{t}=-\Phi_{xxx}+3(\Phi{\cal{D}}\Phi)_{x},\\\
\Psi_{t}=-\Psi_{xxx}+3\big{(}-\Phi{\cal{D}}\Psi+2\Psi{\cal{D}}\Phi\big{)}_{x},\end{array}\right.$
(120)

has a nearly identical form to the twelfth type (118), except for a numerical
factor of “2” in the last term of (120). It is clear that the dark field
$\Psi$ is conserved with the first type of conserved flow
$J=-\Psi_{xx}-3\Phi{\cal{D}}\Psi+6\Psi{\cal{D}}\Phi$.

The fifth symmetry flow of (120) possesses the form

$\left\\{\begin{array}[]{l}\Phi_{\tau}=-\Phi_{xxxxx}+5\big{[}(\Phi_{x}{\cal{D}}\Phi)_{x}+\Phi{\cal{D}}\Phi_{xx}-2\Phi({\cal{D}}\Phi)^{2}\big{]}_{x},\\\
\Psi_{\tau}=5\big{\\{}(2\Psi{\cal{D}}\Phi_{x}-\Phi_{x}{\cal{D}}\Psi)_{x}+2\Psi_{xx}{\cal{D}}\Phi-\Phi{\cal{D}}\Psi_{xx}+4\Phi[\Phi_{x}\Psi+({\cal{D}}\Phi){\cal{D}}\Psi]\\\
\quad\qquad-6({\cal{D}}\Phi)^{2}\Psi\big{\\}}_{x}-\Psi_{xxxxx}.\end{array}\right.$
(121)

Case 14. The fourteenth supersymmetric dark KdV system

$\left\\{\begin{array}[]{l}\Phi_{t}=-\Phi_{xxx}+3(\Phi{\cal{D}}\Phi)_{x},\\\
\Psi_{t}=-4\Psi_{xxx}+{\cal{D}}\big{(}6\Phi\Psi_{x}+3\Phi_{x}\Psi\big{)},\end{array}\right.$
(122)

possesses the first type of conserved density $\Phi$ and the second type of
conserved density $\Psi$ with the conserved flows
$J=-\Phi_{xx}+3\Phi{\cal{D}}\Phi$ and
${\cal{F}}=-4\Psi_{xx}+6\Phi\Psi_{x}+3\Phi_{x}\Psi$, respectively.

The higher order symmetry flows derived from equation (122) can be expressed
as combinations of two distinct types of conservation laws. The first type
corresponds to conservation law formulated in terms of the field $\Phi$, while
the second type pertains to conservation law formulated in terms of the field
$\Psi$, say,

$\left\\{\begin{array}[]{l}\Phi_{\tau}=-\Phi_{xxxxx}+5\big{[}(\Phi_{x}{\cal{D}}\Phi)_{x}+\Phi{\cal{D}}\Phi_{xx}-2\Phi({\cal{D}}\Phi)^{2}\big{]}_{x},\\\
\Psi_{\tau}=5\big{[}{\cal{D}}(3\Phi_{xx}\Psi+8\Phi\Psi_{xx}+4\Phi_{x}\Psi_{x})+4\Phi\Phi_{x}\Psi\big{]}_{x}-16\Psi_{xxxxx}\\\
\qquad\quad+5{\cal{D}}\big{[}3\Phi_{xx}\Psi_{x}-6(\Phi\Psi)_{x}{\cal{D}}\Phi-2\Phi\Phi_{x}{\cal{D}}\Psi\big{]}.\end{array}\right.$
(123)

Case 15. The fifteenth supersymmetric dark KdV system corresponds to

$\left\\{\begin{array}[]{l}\Phi_{t}=-\Phi_{xxx}+3(\Phi{\cal{D}}\Phi)_{x},\\\
\Psi_{t}=-4\Psi_{xxx}+6\Phi{\cal{D}}\Psi_{x}+3\Phi_{x}{\cal{D}}\Psi.\end{array}\right.$
(124)

Regarding the fifth order symmetry flow of equation (124), it can be expressed
as

$\left\\{\begin{array}[]{l}\Phi_{\tau}=-\Phi_{xxxxx}+5\big{[}(\Phi_{x}{\cal{D}}\Phi)_{x}+\Phi{\cal{D}}\Phi_{xx}-2\Phi({\cal{D}}\Phi)^{2}\big{]}_{x},\\\
\Psi_{\tau}=10\Phi\Phi_{x}\Psi_{x}+40\Phi{\cal{D}}\Psi_{xxx}+60\Phi_{x}{\cal{D}}\Psi_{xx}+10(5\Phi_{xx}-3\Phi{\cal{D}}\Phi){\cal{D}}\Psi_{x}\\\
\qquad\quad+5\big{(}3\Phi_{xxx}-2\Phi_{x}{\cal{D}}\Phi-4\Phi{\cal{D}}\Phi_{x}\big{)}{\cal{D}}\Psi-16\Psi_{xxxxx}.\end{array}\right.$
(125)

Case 16. The sixteenth supersymmetric dark KdV system

$\left\\{\begin{array}[]{l}\Phi_{t}=-\Phi_{xxx}+3(\Phi{\cal{D}}\Phi)_{x},\\\
\Psi_{t}=\Phi_{x}{\cal{D}}\Psi+2\Psi_{x}{\cal{D}}\Phi+b({\cal{D}}\Phi_{x})\Psi,\end{array}\right.$
(126)

with an arbitrary constant $b$, can be described as a first order linear
equation for the dark field $\Psi$. Additionally, the higher order symmetry
flows derived from equation (126) can also be expressed as first order linear
equations for the dark field $\Psi$, say

$\left\\{\begin{array}[]{l}\Phi_{\tau}=-\Phi_{xxxxx}+5\big{[}(\Phi_{x}{\cal{D}}\Phi)_{x}+\Phi{\cal{D}}\Phi_{xx}-2\Phi({\cal{D}}\Phi)^{2}\big{]}_{x},\\\
\Psi_{\tau}=2\big{[}{\cal{D}}\Phi_{xx}-3({\cal{D}}\Phi)^{2}+2\Phi\Phi_{x}\big{]}\Psi_{x}+\big{(}\Phi_{xxx}-2\Phi{\cal{D}}\Phi_{x}-4\Phi_{x}{\cal{D}}\Phi\big{)}{\cal{D}}\Psi\\\
\qquad\quad+b[{\cal{D}}\Phi_{xxx}+2\Phi\Phi_{xx}-6({\cal{D}}\Phi)({\cal{D}}\Phi_{x})]\Psi.\end{array}\right.$
(127)

Case 17. The final case corresponds to a degenerate scenario where the system
becomes trivial and completely decoupled

$\left\\{\begin{array}[]{l}\Phi_{t}=-\Phi_{xxx}+3(\Phi{\cal{D}}\Phi)_{x},\\\
\Psi_{t}=-a\Psi_{xxx}\end{array}\right.$ (128)

In this case, the higher order symmetry flows are also trivial and do not
contribute to the system’s behavior

$\left\\{\begin{array}[]{l}\Phi_{t}=-\Phi_{xxx}+3(\Phi{\cal{D}}\Phi)_{x},\\\
\Psi_{t_{2n+1}}=-a_{2n+1}\partial_{x}^{2n+1}\Psi,\ n=1,\ 2,\ \ldots,\
\infty,\end{array}\right.$ (129)

where $a_{2n+1},\ n=1,\ 2,\ \ldots,\ \infty,$ are arbitrary constants.

###  9.2 Supersymmetric integrable dark KdV systems related to the
supersymmetric KdV equation (57)

In this section, we list a comprehensive classification for the supersymmetric
dark KdV system in the form of

$\left\\{\begin{array}[]{l}\Phi_{t}=-\Phi_{xxx}+6\Phi_{x}{\cal{D}}\Phi,\\\
\Psi_{t}=a_{1}\Psi_{xxx}+a_{2}\Phi{\cal{D}}\Psi_{x}+a_{3}({\cal{D}}\Phi_{x})\Psi+a_{4}\Phi_{x}{\cal{D}}\Psi+a_{5}({\cal{D}}\Phi)\Psi_{x}\end{array}\right.$
(130)

with the fifth order symmetry flow

$\left\\{\begin{array}[]{l}\Phi_{\tau}=-\Phi_{xxxxx}+10(\Phi_{xx}{\cal{D}}\Phi)_{x}+10\Phi_{x}{\cal{D}}\Phi_{xx}-30\Phi_{x}({\cal{D}}\Phi)^{2},\\\
\Psi_{\tau}=b_{1}\Psi_{xxxxx}+b_{2}\Phi{\cal{D}}\Psi_{xxx}+b_{3}({\cal{D}}\Phi)\Psi_{xxx}+b_{4}\Phi_{x}{\cal{D}}\Psi_{xx}+b_{5}({\cal{D}}\Phi_{x})\Psi_{xx}+(b_{6}\Phi{\cal{D}}\Phi\\\
\qquad\quad+b_{7}\Phi_{xx}){\cal{D}}\Psi_{x}+[b_{8}({\cal{D}}\Phi)^{2}+b_{9}{\cal{D}}\Phi_{xx}+b_{10}\Phi\Phi_{x}]\Psi_{x}+(b_{11}\Phi_{x}{\cal{D}}\Phi+b_{12}\Phi{\cal{D}}\Phi_{x}\\\
\qquad\quad+b_{13}\Phi_{xxx}){\cal{D}}\Psi+[b_{14}({\cal{D}}\Phi){\cal{D}}\Phi_{x}+b_{15}\Phi\Phi_{xxx}+b_{16}{\cal{D}}\Phi_{xxx}]\Psi,\end{array}\right.$
(131)

where $a_{1},\ a_{2},\ \ldots,\ a_{5}$ and $b_{1},\ b_{2},\ \ldots,\ b_{16}$
are any possible constants.

Through the imposition of consistent conditions, specifically
$\Phi_{t\tau}=\Phi_{\tau t}$ and $\Psi_{t\tau}=\Psi_{\tau t}$, we are able to
ascertain the existence of twelve unique supersymmetric dark KdV systems, each
characterized by a distinct equation structure.

(I.a)
$\qquad\quad\left\\{\begin{array}[]{l}\Phi_{t}=-\Phi_{xxx}+6\Phi_{x}{\cal{D}}\Phi,\\\
\Psi_{t}=-4\Psi_{xxx}+6({\cal{D}}\Phi)\Psi_{x}+3({\cal{D}}\Phi_{x})\Psi,\end{array}\right.$

(I.b)
$\qquad\quad\left\\{\begin{array}[]{l}\Phi_{\tau}=10(\Phi_{xx}u)_{x}+10\Phi_{x}u_{xx}-30\Phi_{x}u^{2}-\Phi_{xxxxx},u\equiv{\cal{D}}\Phi,\\\
\Psi_{\tau}=5(8\Psi_{xx}u+3\Psi
u_{xx}+4\Psi_{x}u_{x}-6u^{2}\Psi)_{x}+15\Psi_{x}u_{xx}+30uu_{x}\Psi-16\Psi_{xxxxx},\end{array}\right.$

which is a supersymmetric form of the usual first type of dark KdV (17).
Applying the supersymmetric derivative ${\cal{D}}$ on the first equation of
(I.a) and making the transformations,

${\cal{D}}\Phi=u,\ \Psi=\zeta v,$ (132)

where $\zeta$ is an arbitrary Grassmann parameter $\zeta$, one can immediately
obtain the usual KdV equation $u_{t}=-u_{xxx}+6uu_{x}$ because of
$\Phi_{x}^{2}=0$ for the fermion field $\Phi$. While the second equation of
(I.a) is just the $v$ equation of (17).

(II.a)
$\qquad\qquad\left\\{\begin{array}[]{l}\Phi_{t}=-\Phi_{xxx}+6\Phi_{x}{\cal{D}}\Phi,\\\
\Psi_{t}=(-\Psi_{xx}+6\Psi{\cal{D}}\Phi)_{x},\end{array}\right.$

(II.b)
$\qquad\qquad\left\\{\begin{array}[]{l}\Phi_{\tau}=-\Phi_{xxxxx}+10(\Phi_{xx}{\cal{D}}\Phi)_{x}+10\Phi_{x}{\cal{D}}\Phi_{xx}-30\Phi_{x}({\cal{D}}\Phi)^{2},\\\
\Psi_{\tau}=[-\Psi_{xxxx}+10(\Psi_{x}{\cal{D}}\Phi)_{x}+10\Psi{\cal{D}}\Phi_{xx}-30({\cal{D}}\Phi)^{2}\Psi]_{x}.\end{array}\right.$

This simple case can be considered as the first kind of supersymmetric form of
the usual symmetry related dark KdV system (22) because the same
transformation (132) makes (II.a) becomes (22).

(III.a)
$\qquad\qquad\left\\{\begin{array}[]{l}\Phi_{t}=-\Phi_{xxx}+6\Phi_{x}{\cal{D}}\Phi,\\\
\Psi_{t}=-\Psi_{xxx}+6{\cal{D}}[({\cal{D}}\Phi){\cal{D}}\Psi],\end{array}\right.$

(III.b)
$\qquad\qquad\left\\{\begin{array}[]{l}\Phi_{\tau}=-\Phi_{xxxxx}+10(\Phi_{xx}{\cal{D}}\Phi)_{x}+10\Phi_{x}{\cal{D}}\Phi_{xx}-30\Phi_{x}({\cal{D}}\Phi)^{2},\\\
\Psi_{\tau}=[-\Psi_{xxx}+10(\Phi_{x}{\cal{D}}\Psi)_{x}+10\Psi_{xx}{\cal{D}}\Phi]_{x}+10{\cal{D}}[\Phi_{xx}\Psi_{x}-3({\cal{D}}\Phi)^{2}{\cal{D}}\Psi].\end{array}\right.$

This case can be considered as the second type of supersymmetric extension of
the usual symmetry related dark KdV system because Eq. (III.a) is related to
(22) by the following transformation

${\cal{D}}\Phi=u,\ {\cal{D}}\Psi=v.$ (133)

(IV.a)
$\qquad\qquad\left\\{\begin{array}[]{l}\Phi_{t}=-\Phi_{xxx}+6\Phi_{x}{\cal{D}}\Phi,\\\
\Psi_{t}=-\Psi_{xxx}+6({\cal{D}}\Phi)\Psi_{x},\end{array}\right.$

(IV.b)
$\qquad\qquad\left\\{\begin{array}[]{l}\Phi_{\tau}=-\Phi_{xxxxx}+10(\Phi_{xx}{\cal{D}}\Phi)_{x}+10\Phi_{x}{\cal{D}}\Phi_{xx}-30\Phi_{x}({\cal{D}}\Phi)^{2},\\\
\Psi_{\tau}=-\Psi_{xxxxx}+10(\Psi_{xx}{\cal{D}}\Phi)_{x}+10[{\cal{D}}\Phi_{xx}-3({\cal{D}}\Phi)^{2}]\Psi_{x}.\end{array}\right.$

By using the transformation (132), the supersymmetric dark KdV system (IV.a)
can be converted to the usual dark KdV system (25) related to the dual
symmetry of the usual KdV equation.

(V.a)
$\qquad\qquad\left\\{\begin{array}[]{l}\Phi_{t}=-\Phi_{xxx}+6\Phi_{x}{\cal{D}}\Phi,\\\
\Psi_{t}=(-\Psi_{xx}+3\Psi{\cal{D}}\Phi)_{x},\end{array}\right.$

(V.b)
$\qquad\qquad\left\\{\begin{array}[]{l}\Phi_{\tau}=-\Phi_{xxxxx}+10(\Phi_{xx}{\cal{D}}\Phi)_{x}+10\Phi_{x}{\cal{D}}\Phi_{xx}-30\Phi_{x}({\cal{D}}\Phi)^{2},\\\
\Psi_{\tau}=[-\Psi_{xxxx}+5(\Psi_{x}{\cal{D}}\Phi)_{x}+5\Psi{\cal{D}}\Phi_{xx}-10\Psi({\cal{D}}\Phi)^{2}]_{x}.\end{array}\right.$

This case is the first type of the supersymmetric dark KdV system related to
the usual strange dark KdV system. By using the special transformation (132),
(V.a) will be changed to the known strange dark KdV system (27) with $a=0$.

(VI.a)
$\qquad\qquad\left\\{\begin{array}[]{l}\Phi_{t}=-\Phi_{xxx}+6\Phi_{x}{\cal{D}}\Phi,\\\
\Psi_{t}=-\Psi_{xxx}+3{\cal{D}}[({\cal{D}}\Phi){\cal{D}}\Psi],\end{array}\right.$

(VI.b)
$\qquad\qquad\left\\{\begin{array}[]{l}\Phi_{\tau}=-\Phi_{xxxxx}+10(\Phi_{xx}{\cal{D}}\Phi)_{x}+10\Phi_{x}{\cal{D}}\Phi_{xx}-30\Phi_{x}({\cal{D}}\Phi)^{2},\\\
\Psi_{\tau}=[-\Psi_{xxxx}+(\Phi_{x}{\cal{D}}\Psi)_{x}+5\Psi_{xx}{\cal{D}}\Phi]_{x}+5{\cal{D}}[\Phi_{xx}\Psi_{x}-({\cal{D}}\Phi)^{2}{\cal{D}}\Psi].\end{array}\right.$

Eq. (VI.a) can be viewed as the second supersymmetric dark KdV extension of
the strange dark KdV system because (27) with $a=0$ can be retrieved from
(VI.a) by using the transformation (133).

(VII.a)
$\qquad\qquad\left\\{\begin{array}[]{l}\Phi_{t}=-\Phi_{xxx}+6\Phi_{x}{\cal{D}}\Phi,\\\
\Psi_{t}=-\Psi_{xxx}+3\Psi_{x}{\cal{D}}\Phi,\end{array}\right.$

(VII.b)
$\qquad\qquad\left\\{\begin{array}[]{l}\Phi_{\tau}=-\Phi_{xxxxx}+10(\Phi_{xx}{\cal{D}}\Phi)_{x}+10\Phi_{x}{\cal{D}}\Phi_{xx}-30\Phi_{x}({\cal{D}}\Phi)^{2},\\\
\Psi_{\tau}=-\Psi_{xxxxx}+5(\Psi_{xx}{\cal{D}}\Phi)_{x}+5[{\cal{D}}\Phi_{xx}-2({\cal{D}}\Phi)^{2}]\Psi_{x}.\end{array}\right.$

The supersymmetric dark KdV system (VII.a) will be reduced back to the usual
dual strange dark KdV system (29) with $a=0$ through the transformation (132).

(VIII.a)
$\qquad\qquad\left\\{\begin{array}[]{l}\Phi_{t}=-\Phi_{xxx}+6\Phi_{x}{\cal{D}}\Phi,\\\
\Psi_{t}=(2\Psi_{xx}-6\Psi{\cal{D}}\Phi)_{x},\end{array}\right.$

(VIII.b)
$\qquad\qquad\left\\{\begin{array}[]{l}\Phi_{\tau}=-\Phi_{xxxxx}+10(\Phi_{xx}{\cal{D}}\Phi)_{x}+10\Phi_{x}{\cal{D}}\Phi_{xx}-30\Phi_{x}({\cal{D}}\Phi)^{2},\\\
\Psi_{\tau}=[4\Psi_{xxxx}-20(\Psi_{x}{\cal{D}}\Phi)_{x}-10\Psi{\cal{D}}\Phi_{xx}+10\Psi({\cal{D}}\Phi)^{2}]_{x}.\end{array}\right.$

This system corresponds to the first type of suppersymmetric case of the
mystery dark KdV system (31) with $a=0$. In fact, the transformation (132) can
lead Eq. (VIII.a) to (31) for $a=0$.

(IX.a)
$\qquad\qquad\left\\{\begin{array}[]{l}\Phi_{t}=-\Phi_{xxx}+6\Phi_{x}{\cal{D}}\Phi,\\\
\Psi_{t}=2\Psi_{xxx}-6{\cal{D}}[({\cal{D}}\Phi){\cal{D}}\Psi],\end{array}\right.$

(IX.b)
$\qquad\qquad\left\\{\begin{array}[]{l}\Phi_{\tau}=-\Phi_{xxxxx}+10(\Phi_{xx}{\cal{D}}\Phi)_{x}+10\Phi_{x}{\cal{D}}\Phi_{xx}-30\Phi_{x}({\cal{D}}\Phi)^{2},\\\
\Psi_{\tau}=[4\Psi_{xxxx}-20(\Psi_{x}{\cal{D}}\Phi)_{x}-10\Phi_{xx}{\cal{D}}\Psi]_{x}+10{\cal{D}}[({\cal{D}}\Psi)({\cal{D}}\Phi)^{2}-\Phi_{xx}\Psi_{x}].\end{array}\right.$

This case can be considered as the second type of suppersymmetric form of the
mystery dark KdV system (31). If we apply the transformation (133), then Eq.
(IX.a) becomes (31) with $a=0$.

(X.a)
$\qquad\qquad\left\\{\begin{array}[]{l}\Phi_{t}=-\Phi_{xxx}+6\Phi_{x}{\cal{D}}\Phi,\\\
\Psi_{t}=2\Psi_{xxx}-6({\cal{D}}\Phi)\Psi_{x},\end{array}\right.$

(X.b)
$\qquad\qquad\left\\{\begin{array}[]{l}\Phi_{\tau}=-\Phi_{xxxxx}+10(\Phi_{xx}{\cal{D}}\Phi)_{x}+10\Phi_{x}{\cal{D}}\Phi_{xx}-30\Phi_{x}({\cal{D}}\Phi)^{2},\\\
\Psi_{\tau}=4\Psi_{xxxxx}-20(\Psi_{xx}{\cal{D}}\Phi)_{x}-10\Psi_{x}[{\cal{D}}\Phi_{xx}-({\cal{D}}\Phi)^{2}].\end{array}\right.$

The usual dual mystery dark KdV system (33) can be obtained from the
supersymmetric dark KdV system (X.a) with $a=b=0$ by utilizing the
transformation (132).

(XI.a)
$\qquad\qquad\left\\{\begin{array}[]{l}\Phi_{t}=-\Phi_{xxx}+6\Phi_{x}{\cal{D}}\Phi,\\\
\Psi_{t}=a\Psi{\cal{D}}\Psi_{x}+2\Psi_{x}{\cal{D}}\Phi,\end{array}\right.$

(XI.b)
$\qquad\qquad\left\\{\begin{array}[]{l}\Phi_{\tau}=-\Phi_{xxxxx}+10(\Phi_{xx}{\cal{D}}\Phi)_{x}+10\Phi_{x}{\cal{D}}\Phi_{xx}-30\Phi_{x}({\cal{D}}\Phi)^{2},\\\
\Psi_{\tau}=2[{\cal{D}}\Phi_{xx}-3({\cal{D}}\Phi)^{2}]\Psi_{x}+a[{\cal{D}}\Phi_{xxx}-6({\cal{D}}\Phi){\cal{D}}\Phi_{x}]\Psi.\end{array}\right.$

The usual ninth type dark KdV system (37) with (38) and $b=0$ is a special
case of the supersymmetric dark KdV system (XI.a) via the transformation
(132).

(XII.a)
$\qquad\qquad\left\\{\begin{array}[]{l}\Phi_{t}=-\Phi_{xxx}+6\Phi_{x}{\cal{D}}\Phi,\\\
\Psi_{t}=a\Psi_{xxx},\end{array}\right.$

(XII.b)
$\qquad\qquad\left\\{\begin{array}[]{l}\Phi_{\tau}=-\Phi_{xxxxx}+10(\Phi_{xx}{\cal{D}}\Phi)_{x}+10\Phi_{x}{\cal{D}}\Phi_{xx}-30\Phi_{x}({\cal{D}}\Phi)^{2},\\\
\Psi_{\tau}=b\Psi_{xxxxx}.\end{array}\right.$

This supersymmetric dark KdV extension exhibits a trivial and fully decoupled
nature, sharing similarities with the standard decoupled dark KdV system
described by equation (35) when $a$ and $b$ are both set to zero.

Within this subsection, it has been observed that every homogeneous dark KdV
system documented in Section 2 possesses, at minimum, one supersymmetric
extension. Notably, for the dark KdV systems pertaining to the symmetry case
denoted by equation (22), the strange case represented by equation (27), and
the mysterious case characterized by equation (31), two distinct categories of
supersymmetric dark KdV extensions exist.

##  10 Concluding remarks

In summary, a plethora of dark equation systems can be derived through a
variety of methodologies, encompassing the bosonization technique applied to
supersymmetric integrable systems and super-integrable models, the dark
parameter expansion method, and the decomposition method of higher-dimensional
integrable systems. These methodologies offer powerful avenues for
investigating the intricate nature of dark equations, unveiling their
properties. The existence of an infinite number of such systems underscores
the profound complexity and richness inherent in the realm of dark equations,
fostering continuous exploration and discovery in this cutting-edge field.
Such advancements hold great promise for advancing our understanding of
fundamental physics and may have far-reaching implications in various
scientific disciplines.
The dark equations elucidated in this paper exhibit remarkable integrability
properties, primarily due to the presence of higher order symmetries. Indeed,
the systems under investigation may exhibit additional integrable properties.
These properties encompass a wide range of integrability aspects, including
Lax integrability, recursion operators, bi-Hamiltonian structures, Bäcklund
and Darboux transformations, conservation laws, and the Painlevé property,
among others. For instance, the dark KdV system, as represented by equation
(35), possesses a recursion operator

$\phi=\left(\begin{array}[]{cc}\phi_{1}&0\\\
\phi_{2}&\gamma_{1}\partial_{x}^{2}\end{array}\right)$

with $\phi_{i}=c_{i}\partial_{x}^{2}-4u-2u_{x}\partial_{x}^{-1},\ i=1,\ 2,\
c_{1}=1,\ c_{2}=1-\gamma_{1}$.

Under the framework of Lax pairs and bi-Hamiltonian integrability [6], the Lax
pair
($\hat{\cal{L}}\equiv-4\partial_{x}^{3}+6\Phi{\cal{D}}^{3}+3\Phi_{x}{\cal{D}}$)

$\displaystyle\left(\begin{array}[]{cc}\partial_{x}^{2}-\Phi{\cal{D}}&0\\\
-\Phi_{1}{\cal{D}}-q&\partial_{x}^{2}-\Phi{\cal{D}}\end{array}\right)\left(\begin{array}[]{c}\Psi_{1}\\\
\Psi_{2}\end{array}\right)=\left(\begin{array}[]{cc}\lambda&0\\\
0&\lambda\end{array}\right)\left(\begin{array}[]{c}\Psi_{1}\\\
\Psi_{2}\end{array}\right),$ (142)
$\displaystyle\left(\begin{array}[]{c}\Psi_{1}\\\
\Psi_{2}\end{array}\right)_{t}=\left(\begin{array}[]{cc}\hat{\cal{L}}&0\\\
6\Phi_{1}{\cal{D}}^{3}+6r{\cal{D}}^{2}+3\Phi_{1x}{\cal{D}}+3s&\hat{\cal{L}}-3v\end{array}\right)\left(\begin{array}[]{c}\Psi_{1}\\\
\Psi_{2}\end{array}\right),$ (149)

defines a supersymmetric Lax integrable dark KdV system

$\displaystyle\Phi_{t}=(3\Phi{\cal{D}}\Phi-\Phi_{xx})_{x},$
$\displaystyle\Phi_{1t}=(3\Phi{\cal{D}}\Phi_{1}+\Phi_{1}{\cal{D}}\Phi+6r\Phi-\Phi_{1xx})_{x}-3v\Phi_{1}$
$\displaystyle
q_{t}=\left[(3\Phi_{1}\Phi)_{x}+3\Phi{\cal{D}}r-q_{xx}\right]_{x}+3\Phi{\cal{D}}q_{x}-3vq-3\Phi{\cal{D}}s,$
(150)

where $\Phi$ and $\Phi_{1}$ are two fermionic super-fields and $q$ is a
bosonic super-field while the other three bosonic potential fields $\\{v,\ r,\
s\\}$ are related to $\Phi,\ \Phi_{1}$ and $q$ by

$v_{x}=\Phi\Phi_{x},\ r_{x}=q_{x}+\Phi\Phi_{1},\
s_{x}=(\Phi_{1}\Phi+q_{x})_{x}-\Phi{\cal{D}}(q-r).$

According to the findings presented in this paper, the notion of dark
equations extends beyond a mere conceptual analogy to dark matter and dark
energy. In fact, there exists a tangible connection between dark equations and
dark matter, as various dark equations can be directly associated with
supersymmetric systems. These systems, in turn, are intricately linked to
supergravity and dark matter phenomena. The results of this study indicate the
presence of multiple dark fields within a single supersymmetric framework.
This revelation underscores the complexity and richness of dark equations
within the context of supersymmetry, shedding light on the potential interplay
between these equations and the enigmatic nature of dark matter. The presence
of such diverse integrable features highlights the profound mathematical
structure inherent in these systems. The identification and exploration of
these properties pave the way for a deeper understanding of the intricate
dynamics and profound connections associated with dark equations, unraveling
the intricate interplay between various integrable structures and shedding
light on the fundamental nature of these systems.

## Acknowledgments

The work was sponsored by the National Natural Science Foundations of China
(Nos.12235007, 11975131). The author is indebt to thank Profs. X. Z. Hao, Q.
P. Liu, R. X. Yao, M. Jia, X. B. Hu and K. Tian for their helpful discussions.

## References

  * [1] M. Ozer and M. O. Taha, A Possible Solution to the Main Cosmological Problems, Phys. Lett. B 171 (1986) 363.
  * [2] C. R. Argüelles, E. A. Becerra-Vergara, J. A. Rueda, and R. Ruffini, Fermionic dark matter: physics, astrophysics, and cosmology, Universe 9 (2023) 197.
  * [3] D. Wang, M. Koussour, A. Malik, N. Myrzakulov and G. Mustafa, Observational constraints on a logarithmic scalar field dark energy model and black hole mass evolution in the universe, Euro. Phys. J. C, 83 (2023) 670.
  * [4] A. G. Riess et al., Observational evidence from supernovae for an accelerating Universe and a cosmological constant, Astron. J. 116 (1998) 1009.
  * [5] S. Perlmutter et al., Measurements of $\Omega$ and $\Lambda$ from 42 high-redshift supernovae, Astrophys. J. 517 (1999) 565.
  * [6] B. A. Kupershmidt, Dark equations, J. Nonl. Math. Phys. 8 (2001)363.
  * [7] W. X. Ma and B. Fuchssteiner, The bi-Hamiltonian structure of the perturbation equations of the KdV hierarchy, Phys. Lett. A 213 (1996) 49.
  * [8] Y. Geyer and L. Mason, Superstring loop amplitudes from the field theory limit, Phys. Rev. Lett., 127 (2021) 211603.
  * [9] J. L. Feng, J. F. Grivaz and J. Nachtman, Searches for Supersymmetry at High-Energy Colliders, Rev. Mod. Phys. 82 (2010) 699.
  * [10] S. A. Abel, S. Sarkar and I. B. Whittingham, Neutralino dark matter in a class of unified theories, Nucl. Phys. B, 392 (1993) 83.
  * [11] V. V. Flambaum, X. Liu, I. Samsonov, L. Wu and B. Zhu, Probing supersymmetry breaking scale with atomic clocks, Nucl. Phys. B 993 (2023) 116260.
  * [12] X. N. Gao and S. Y. Lou, Bosonization of supersymmetric KdV equation, Phys. Lett. B 707 (2012) 209.
  * [13] B. A. Kupershmidt, A super Korteweg-de Vries equation: an integrable system, Phys. Lett. 102A (1984) 213.
  * [14] X. Z. Hao and S. Y. Lou, Special decompositions and linear superpositions of nonlinear systems: BKP and dispersionless BKP equations, Math. Meth. Appl. Sci. 45 (2022) 5774.
  * [15] L. Peng, X. D. Yang and S. Y. Lou, Integrable KP Coupling and Its Exact Solution, Commun. Theor. Phys. 58 (2012) 1.
  * [16] S. Majid and M. J. Rodräguez-Plaza, Random walk and the heat equation on superspace and anyspace, J. Math. Phys. 35 (1994) 3753.
  * [17] S. Y. Lou, Ren-integrable and ren-symmetric integrable systems, arXiv: 2305. 12388v1 [nlin.SI] (2023).
  * [18] S. Y. Lou and B. F. Feng, Symmetries and integrable systems, Fundamental Research, (2023) DOI.10.1016/j.fmre.2023.11.008
  * [19] K. Tian and Q. P. Liu, Supersymmetric fifth order evolution equations, AIP Conf. Proc. 1212 (2010) 81.
  * [20] X. N. Gao, S. Y. Lou and X. Y. Tang, Bosonization, singularity analysis, nonlocal symmetry reductions and exact solutions of supersymmetric KdV equation, JHEP 5 (2013) 029.
  * [21] B. A. Kupershmidt, Super-integrable systems, Proc. Nat. Acad. Sci. USA, 81 (1984) 6562.
  * [22] B. A. Kupershmidt, Bosons and fermions interacting integrably with the Korteweg-de Vries field, J. Phys. A 17 (1984) L869.
  * [23] X. Z. Hao and S. Y. Lou, Decomposition solutions and Bäcklund transformations of the BKP and CKP equations, arXiv: 2210.00542v1 [nlin.SI] (2022).
  * [24] P. Mathieu, Supersymmetric extension of the Korteweg-de Vries equation, J. Math. Phys 29 (1988) 2499.