Generalized Cross-Curvature Solitons of 3D Lorentzian Lie Groups

Abstract

We investigate left-invariant generalized cross-curvature solitons on simply connected three-dimensional Lorentzian Lie groups. Working with the assumption that the contravariant tensor $P_{ij}$ (defined from the Ricci tensor and scalar curvature) is invertible, we derive the algebraic soliton equations for left-invariant metrics and classify all left-invariant generalized cross-curvature solitons (for the generalized equation ${\mathcal{L}}_{X}g+\lambda g=2h+2\rho Rg$) on the standard 3D Lorentzian Lie algebra types (unimodular Types Ia, Ib, II, and III and non-unimodular Types IV.1, IV.2, and IV.3). For each Lie algebra type, we state the necessary and sufficient algebraic conditions on the structure constants, provide explicit formulas for the soliton vector fields X (when they exist), and compute the soliton parameter $\lambda$ in terms of the structure constants and the parameter $\rho$. Our results include several existence families, explicit nonexistence results (notably for Type Ib and Type IV.3), and consequences linking the existence of left-invariant solitons with local conformal flatness in certain cases. The classification yields new explicit homogeneous generalized cross-curvature solitons in the Lorentzian setting and clarifies how the parameter $\rho$ modifies the algebraic constraints. Examples and brief geometric remarks are provided.

1. Introduction

Geometric flows constitute an essential methodology for continuously transforming a given metric on a manifold towards a canonical or more regular geometric configuration. Prominent examples of such geometric flows, fundamentally significant in both theoretical physics and differential geometry, encompass the cross-curvature (CC) flow, Ricci-harmonic flow, mean curvature flow, Yamabe flow, and Ricci flow. A significant subset of solutions within these flows are the so-called solitons, which characterize self-similar solutions evolving under the flow dynamics. Ricci solitons, initially introduced by R. Hamilton in [1], satisfy the defining relation

$${\displaystyle \frac{1}{2}}{\mathcal{L}}_{X}g+\mathrm{Ric}=\lambda g,$$

and naturally extend the concept of Einstein metrics. Since their inception, this notion has been broadened to encompass solitons arising from various other geometric flow equations. Here, the term ‘curvature’ refers exclusively to the intrinsic (pseudo-)Riemannian curvature of three-dimensional manifolds; we do not consider extrinsic curvature nor make direct physical interpretations from General Relativity.

This study is devoted to the analysis of generalized CC solitons residing on 3D Lorentzian homogeneous manifolds. Such manifolds are characterized either by local symmetry or by a local Lie group structure equipped with a Lorentzian metric that remains invariant under left translations. Our primary objective is to classify such solitons and analyze their intrinsic geometric characteristics within the Lorentzian framework.

Consider a smooth 3D manifold $(M,g)$ equipped with a metric g. Define the symmetric $(0,2)$-tensor

$$P_{ij}=R_{ij}-{\displaystyle \frac{1}{2}}R{g}_{ij},$$

(1)

where $R_{ij}$ denotes the Ricci tensor components and R is the scalar curvature derived from g. By raising indices through the inverse metric tensor $g^{ij}$, the corresponding $(2,0)$-tensor is given by

$$P^{ij}=g^{ik}{g}^{jl}{R}_{kl}-{\displaystyle \frac{1}{2}}R{g}^{ij}.$$

Assuming that the matrix $\left(P^{ij}\right)$ is invertible with inverse denoted by $V_{ij}$, the CC tensor h is introduced by

$$h_{ij}={\displaystyle \frac{det\left(P^{kl}\right)}{det\left(g^{kl}\right)}}{V}_{ij}.$$

A pseudo-Riemannian manifold $(M,g)$ of dimension three is termed a cross-curvature soliton (CCS) if a smooth vector field X and a constant scalar $\lambda$ exist, satisfying

$${\mathcal{L}}_{X}g+\lambda g=2h,$$

(2)

where ${\mathcal{L}}_X$ is the Lie derivative of the metric g, and h is as above. The cross-curvature flow (CCF), formulated by B. Chow and R. S. Hamilton [2], applies to three-dimensional Riemannian manifolds and is defined by the differential equation

$${\displaystyle \frac{\partial g}{\partial t}}=-2ϵh,g\left(0\right)=g_0,$$

(3)

where the tensor h denotes the cross-curvature tensor, and the parameter $ϵ=\pm 1$ reflects the sign of the sectional curvature of $g_0$. Such solitons characterize self-similar solutions to the CC flow and play a role analogous to that of Ricci solitons in the context of Ricci flow, as studied in [3,4,5,6].

The tensor $P_{ij}$ introduced in Equation (1) can be interpreted as a curvature-derived analogue of an energy–momentum tensor. In three dimensions, all curvature information is encoded in the Ricci tensor $R_{ij}$ (or equivalently the Schouten tensor). From this perspective, $P_{ij}$ provides a natural symmetric two-tensor constructed purely from intrinsic curvature data, serving as the appropriate analogue of a stress–energy-type tensor. The generalized soliton Equation (2) should therefore be regarded as a curvature-driven ansatz, motivated by the same algebraic principles that underlie stress–energy constructions in geometric flows and gravity. The trace of the tensor P on the surfaces is zero.

In our analysis, we assume the invertibility of the tensor $P_{ij}$. This assumption is natural for most left-invariant metrics under consideration, and it plays an analytic role in reducing the soliton system to a tractable set of relations. However, for completeness, we now explicitly list the degenerate cases in Section 2 where $P_{ij}$ fails to be invertible. Such exceptional cases are either excluded from the present analysis or treated separately with modified soliton equations. The CC flow on three-dimensional locally homogeneous Riemannian spaces with non-negative curvature constraints was studied by Cao and collaborators [7,8], who analyzed the long-term behavior of the flow. Further contributions to this field can be found in [9,10,11,12].

Extending this notion, a generalized cross-curvature soliton is defined by the equation

$${\mathcal{L}}_{X}g+\lambda g=2h+2\rho Rg,$$

(4)

where $\rho$ is a real parameter and R denotes the scalar curvature. This generalization was introduced by Azami [13] to describe solitons of the generalized CC flow. The soliton is called expanding, steady, or shrinking depending on whether $\lambda$ is positive, zero, or negative, respectively. If the vector field X is Killing, i.e., ${\mathcal{L}}_{X}g=0$, then the soliton is considered trivial.

Classical soliton (kink) solutions originally arise in the study of solitary waves in nonlinear field theories and are often constructed as finite-energy solutions of the (Lorentzian) field equations. The concept of a “soliton” was first introduced by Kruskal and Zabusky to describe the nature of solitary waves [14]. Since then, soliton theory has been extensively developed and found applications in fields such as fluid dynamics, elementary particle physics, and condensed matter physics [15]. A relatively complete mathematical and physical theory of solitons has now emerged, demonstrating close connections with modern physics. On the other hand, symmetric metrics are often employed to simplify the classification of solutions to Einstein’s field equations. In this context, solitons represent an important type of symmetry related to the geometric flow of spacetime. Hamilton [1] introduced the Ricci flow and the Yamabe flow, along with their associated solitons, which have been instrumental in understanding the kinematics of geometric structures.

In semiclassical quantum mechanics and quantum field theory, it is standard to perform a Wick rotation $t\mapsto i\tau$ to Euclidean time; under this transformation, certain time-dependent Lorentzian solutions become static or finite-action solutions of the Euclidean equations. Such finite-action Euclidean solutions are commonly referred to as instantons or pseudo-particles and are used to evaluate tunneling amplitudes and semiclassical contributions to path integrals (for reviews, see, e.g., [16,17]).

In the present work, when we refer to soliton or kink solutions, we mean (depending on context) either the original Lorentzian solitary-wave solutions or their Euclidean counterparts used in semiclassical analysis. Likewise, the phrase time-dependent solution in the Euclidean formulation should be read as a $\tau$-dependent (Euclidean-time dependent) solution. The metrics generated by the cross-curvature flow depend on time ttt, satisfy Equation (3), and their special cases are the cross-curvature solitons.

The foundational results on the generalized CC flow, including its short-time existence, were established in [13].

In recent developments, the study of various geometric solitons on locally homogeneous manifolds has attracted considerable attention. Previous studies have established that no non-trivial homogeneous Ricci solitons arise on Lie groups of dimension four or lower when these groups are equipped with left-invariant Riemannian metrics (refer to [18,19,20,21]). Nevertheless, explicit examples of three-dimensional homogeneous Ricci solitons in Riemannian settings have been constructed in [22,23]. Onda [24] extended a key result originally established by Lauret [25], who demonstrated that Ricci solitons of algebraic type on Lie groups equipped with left-invariant (LI) Riemannian metrics are inherently homogeneous. This foundational insight was later adapted to the broader setting of pseudo-Riemannian geometry, revealing its applicability beyond the Riemannian case. Moreover, Calvaruso and Fino [26] explored Ricci solitons on homogeneous non-reductive four-dimensional spaces.

Regarding CCSs, the classification of LI solutions on three-dimensional Lorentzian Lie groups (3DLLGs) has been presented in [9]. For more information on Ricci solitons in homogeneous geometries, the reader may refer to [27,28].

Recent interest in geometric evolution equations beyond the Ricci flow has motivated the study of cross-curvature (CC) flows and their solitons as natural pseudo-Riemannian analogues of Ricci solitons. In particular, extending the CC soliton framework to Lorentzian signature and introducing a one-parameter generalization involving the parameter $\rho$ yields a richer algebraic structure and new families of homogeneous models to analyze. Left-invariant metrics on three-dimensional Lie groups provide a tractable setting for a complete algebraic classification, allowing for the explicit computation of curvature tensors and reduction in the soliton condition to solvable polynomial systems. Recent work by Azami et al. (2024) [9] on cross-curvature solitons of three-dimensional Lorentzian Lie groups provides both a precedent and concrete examples that motivate the present generalization. Our introduction of the parameter $\rho$ unifies and extends previously studied cases, clarifies the role of conformal flatness in the existence theory, and produces new existence and nonexistence results specific to the Lorentzian setting. These homogeneous models therefore serve as useful laboratories for testing conjectures about existence, uniqueness, and stability for generalized CC-flows, and they highlight phenomena absent in the Riemannian category.

The organization of this paper is as follows: Section 2 recalls necessary background on 3DLLGs utilized in the subsequent classification. Section 3 contains the main results along with their proofs.

2. 3DLLGs

We provide a brief summary of the classification of 3DLLGs, encompassing both unimodular and non-unimodular types. According to the classification outlined in [29], any three-dimensional Lorentzian homogeneous manifold that is complete and simply connected must either exhibit local symmetry or be isometrically equivalent to a Lie group equipped with an LI Lorentzian metric.

2.1. Unimodular Lie Groups

Let us consider the Lorentzian vector space ${\mathbb{R}}_1^3$ equipped with a metric of signature $(+,+,-)$ and fix an orthonormal basis $\{e_1,e_2,e_3\}$. The Lorentzian cross product × on this space is defined by the relations

$$e_1\times e_2=e_3,e_2\times e_3=e_1,e_3\times e_1=e_2,$$

which correspond to the para-quaternionic multiplication structure.

The Lie bracket on the Lie algebra $\mathfrak{g}$ associated with the Lie group can be represented in terms of a linear map $L:{\mathbb{R}}_1^3\to {\mathbb{R}}_1^3$ as

$$[Z,Y]=L(Z\times Y),Z,Y\in \mathfrak{g}.$$

Lie algebra $\mathfrak{g}$ is classified as unimodular precisely when the linear endomorphism L is self-adjoint relative to the Lorentzian scalar product; if this condition fails, $\mathfrak{g}$ is deemed non-unimodular [30]. The classification of L according to its algebraic form yields four main unimodular Lie algebra types [31].

Type Ia.

Assume that L is diagonalizable relative to an orthonormal basis $\{e_1,e_2,e_3\}$ with signature $(+,+,-)$, with eigenvalues $\alpha ,\beta ,\gamma$. Then, the Lie algebra structure is given by the brackets

$$[e_2,e_3]=\alpha e_1,[e_1,e_3]=-\beta e_2,[e_1,e_2]=-\gamma e_3.$$

The Levi–Civita connection ∇ satisfies

$${\nabla }_{e_i}{e}_j={\displaystyle \frac{1}{2}}\left(\begin{array}{ccc}0& (\alpha -\beta -\gamma )e_3& (\alpha -\beta -\gamma )e_2\\ (\alpha -\beta +\gamma )e_3& 0& (\alpha -\beta +\gamma )e_1\\ (\alpha +\beta -\gamma )e_2& -(\alpha +\beta -\gamma )e_1& 0\end{array}\right),$$

where the entry in position $(i,j)$ corresponds to the vector ${\nabla }_{e_i}{e}_j$.

By applying the curvature operator formula

$$R(V,Y)={\nabla }_{[V,Y]}-[{\nabla }_V,{\nabla }_Y],$$

we obtain the nonzero terms of the curvature tensor as

$$\begin{array}{cc}\hfill R_{2332}& ={\displaystyle \frac{1}{4}}\left(2\beta \gamma -2\alpha \gamma -2\alpha \beta -{\gamma }^2-{\beta }^2+3{\alpha }^2\right),\hfill \\ \hfill R_{1313}& ={\displaystyle \frac{1}{4}}\left(2\beta \gamma -2\alpha \gamma +2\alpha \beta +{\gamma }^2-3{\beta }^2+{\alpha }^2\right),\hfill \\ \hfill R_{1221}& ={\displaystyle \frac{1}{4}}\left(2\beta \gamma +2\alpha \gamma -2\alpha \beta -3{\gamma }^2+{\beta }^2+{\alpha }^2\right).\hfill \end{array}$$

The Ricci tensor matrix takes the diagonal form

$$R_{11}=-{\displaystyle \frac{1}{2}}\left({\alpha }^2-{(\gamma -\beta )}^2\right),R_{22}=-{\displaystyle \frac{1}{2}}\left({\beta }^2-{(\gamma -\alpha )}^2\right),R_{33}={\displaystyle \frac{1}{2}}\left({\gamma }^2-{(\beta -\alpha )}^2\right).$$

with all other entries equal to zero. Using the formula

$${\mathcal{L}}_{X}g(Y,Z)=g({\nabla }_{Y}X,Z)+g(Y,{\nabla }_{Z}X),$$

for an LI vector field $X={\sum }_{i=1}^3{x}_i{e}_i$, ${\mathcal{L}}_{X}g$ can be represented as

$$\left({\mathcal{L}}_{X}g\right)=\left(\begin{array}{ccc}0& (\alpha -\beta )x_3& (\gamma -\alpha )x_2\\ (\alpha -\gamma )x_3& 0& (\beta -\gamma )x_1\\ (\gamma -\alpha )x_2& (\beta -\gamma )x_1& 0\end{array}\right).$$

The scalar curvature is explicitly computed as

$$R={\displaystyle \frac{1}{2}}\left({\alpha }^2+{\beta }^2+{\gamma }^2-2\beta \gamma -2\alpha \gamma -2\alpha \beta \right).$$

The components of the tensor $P^{ij}$ have only diagonal entries given by

$$\begin{array}{cc}\hfill P^{11}& =-{\displaystyle \frac{1}{4}}\left(2\beta \gamma -2\alpha \gamma -2\alpha \beta -{\gamma }^2-{\beta }^2+3{\alpha }^2\right),\hfill \\ \hfill P^{22}& ={\displaystyle \frac{1}{4}}\left(2\beta \gamma -2\alpha \gamma +2\alpha \beta +{\gamma }^2-3{\beta }^2+{\alpha }^2\right),\hfill \\ \hfill P^{33}& =-{\displaystyle \frac{1}{4}}\left(2\beta \gamma +2\alpha \gamma -2\alpha \beta -3{\gamma }^2+{\beta }^2+{\alpha }^2\right).\hfill \end{array}$$

with all off-diagonal terms being zero. We will assume $\left(P^{ij}\right)$ is invertible in what follows. The CC tensor h is diagonal with entries

$$\begin{array}{cc}\hfill h_{11}& =-{\displaystyle \frac{1}{16}}\left(2\beta \gamma -2\alpha \gamma +2\alpha \beta +{\gamma }^2-3{\beta }^2+{\alpha }^2\right)\left(-2\beta \gamma -2\alpha \gamma +2\alpha \beta +3{\gamma }^2-{\beta }^2-{\alpha }^2\right),\hfill \\ \hfill h_{22}& =-{\displaystyle \frac{1}{16}}\left(2\beta \gamma +2\alpha \gamma -2\alpha \beta +{\gamma }^2+{\beta }^2-3{\alpha }^2\right)\left(-2\beta \gamma -2\alpha \gamma +2\alpha \beta +3{\gamma }^2-{\beta }^2-{\alpha }^2\right),\hfill \\ \hfill h_{33}& =-{\displaystyle \frac{1}{16}}\left(2\beta \gamma +2\alpha \gamma -2\alpha \beta +{\gamma }^2+{\beta }^2-3{\alpha }^2\right)\left(2\beta \gamma -2\alpha \gamma +2\alpha \beta +{\gamma }^2-3{\beta }^2+{\alpha }^2\right).\hfill \end{array}$$

Type Ib.

Let the linear map L on ${\mathbb{R}}_1^3$ have the matrix representation

$$L=\left(\begin{array}{ccc}\alpha & 0& 0\\ 0& \gamma & -\beta \\ 0& \beta & \gamma \end{array}\right),$$

with $\beta \ne 0$. Then the Lie brackets defining ${\mathfrak{g}}_{Ib}$ are

$$[e_2,e_3]=\alpha e_1,[e_1,e_3]=-\beta e_3-\gamma e_2,[e_1,e_2]=-\gamma e_3+\beta e_2.$$

Then,

$$\left({\nabla }_{e_i}{e}_j\right)=\left(\begin{array}{ccc}0& {\displaystyle \frac{1}{2}}(\alpha -2\gamma )e_3& {\displaystyle \frac{1}{2}}(\alpha -2\gamma )e_2\\ -\beta e_2+{\displaystyle \frac{\alpha }{2}}{e}_3& \beta e_1& {\displaystyle \frac{\alpha }{2}}{e}_1\\ \beta e_3+{\displaystyle \frac{\alpha }{2}}{e}_2& -{\displaystyle \frac{\alpha }{2}}{e}_1& \beta e_1\end{array}\right).$$

Nonzero curvature components are given by

$$R_{1221}=R_{1313}={\displaystyle \frac{1}{4}}({\alpha }^2+4{\beta }^2),R_{2332}={\displaystyle \frac{3}{4}}{\alpha }^2-\alpha \gamma +{\beta }^2,R_{1231}=-2\beta \gamma +\beta \alpha .$$

The Ricci tensor is represented by

$$Ric=\left(\begin{array}{ccc}-{\displaystyle \frac{1}{2}}({\alpha }^2+4{\beta }^2)& 0& 0\\ 0& {\displaystyle \frac{1}{2}}\alpha (\alpha -2\gamma )& -\beta (\alpha -2\gamma )\\ 0& -\beta (\alpha -2\gamma )& -{\displaystyle \frac{1}{2}}\alpha (\alpha -2\gamma )\end{array}\right).$$

Consider an LI vector field $X={\sum }_{i=1}^3{x}_i{e}_i$. The operator ${\mathcal{L}}_{X}g$ is obtained by

$${\mathcal{L}}_{X}g=\left(\begin{array}{ccc}0& \beta x_2+(\alpha -\gamma )x_3& \beta x_3+(\gamma -\alpha )x_2\\ \beta x_2+(\alpha -\gamma )x_3& -2\beta x_1& 0\\ \beta x_3+(\gamma -\alpha )x_2& 0& -2\beta x_1\end{array}\right).$$

The scalar curvature simplifies to

$$R={\displaystyle \frac{1}{2}}\left({\alpha }^2-4{\beta }^2-4\alpha \gamma \right).$$

Tensor $P^{ij}$ takes the form

$$P^{ij}=\left(\begin{array}{ccc}{\displaystyle \frac{1}{4}}(-3{\alpha }^2-4{\beta }^2+4\alpha \gamma )& 0& 0\\ 0& {\displaystyle \frac{1}{4}}{\alpha }^2+{\beta }^2& 2\beta ({\displaystyle \frac{1}{2}}\alpha -\gamma )\\ 0& 2\beta ({\displaystyle \frac{1}{2}}\alpha -\gamma )& -({\displaystyle \frac{1}{4}}{\alpha }^2+{\beta }^2)\end{array}\right).$$

Setting

$$A_1=-\left({\displaystyle \frac{1}{16}}{({\alpha }^2+4{\beta }^2)}^2+{\beta }^2{(\alpha -2\gamma )}^2\right),$$

the inverse matrix $\left(V_{ij}\right)={\left(P^{ij}\right)}^{-1}$ is given by

$$V_{ij}={\displaystyle \frac{1}{A_1}}\left(\begin{array}{ccc}-4{\displaystyle \frac{\frac{1}{16}{({\alpha }^2+4{\beta }^2)}^2+{\beta }^2{(\alpha -2\gamma )}^2}{-3{\alpha }^2-4{\beta }^2+4\alpha \gamma }}& 0& 0\\ 0& -{\displaystyle \frac{1}{4}}({\alpha }^2+4{\beta }^2)& -\beta \alpha +2\beta \gamma \\ 0& -\beta \alpha +2\beta \gamma & {\displaystyle \frac{1}{4}}({\alpha }^2+4{\beta }^2)\end{array}\right).$$

Finally, the CC tensor has components

$$\left(h_{ij}\right)=-{\displaystyle \frac{1}{4}}(-3{\alpha }^2-4{\beta }^2+4\alpha \gamma )·V_{ij}.$$

Type II.

Let us consider a linear transformation L acting on a Lorentzian vector space with an orthonormal basis $\{e_1,e_2,e_3\}$, where the metric signature is $(+,+,-)$, and L exhibits a repeated eigenvalue in its minimal polynomial. The matrix form of L is explicitly given by

$$L=\left(\begin{array}{ccc}\alpha & 0& 0\\ 0& \frac{1}{2}+\beta & -\frac{1}{2}\\ 0& \frac{1}{2}& -\frac{1}{2}+\beta \end{array}\right).$$

The Lie algebra ${\mathfrak{g}}_{II}$ associated with this structure is determined by the following commutation relations:

$$[e_2,e_3]=\alpha e_1,[e_1,e_3]=-\frac{1}{2}{e}_3-\left(\beta +\frac{1}{2}\right)e_2,[e_1,e_2]=\frac{1}{2}{e}_2-\left(-\frac{1}{2}+\beta \right)e_3.$$

The Levi–Civita connection ∇ on this Lie algebra admits the following representation:

$$\left({\nabla }_{e_i}{e}_j\right)=\left(\begin{array}{ccc}0& \frac{1}{2}(\alpha -2\beta )e_3& \frac{1}{2}(\alpha -2\beta )e_2\\ -\frac{1}{2}{e}_2+\frac{1}{2}(\alpha -1)e_3& \frac{1}{2}{e}_1& \frac{1}{2}(\alpha -1)e_1\\ \frac{1}{2}(\alpha +1)e_2+\frac{1}{2}{e}_3& -\frac{1}{2}(\alpha +1)e_1& \frac{1}{2}{e}_1\end{array}\right).$$

Nonzero curvature components relative to the basis $\{e_1,e_2,e_3\}$ are expressed as

$$\begin{array}{cccc}\hfill R_{1231}& =\frac{1}{2}\alpha -\beta ,\hfill & \hfill & R_{2332}=-{\displaystyle \frac{1}{4}}\alpha (4\beta -3\alpha ),\hfill \\ \hfill R_{1313}& ={\displaystyle \frac{1}{4}}\left(-4\beta +2\alpha +{\alpha }^2\right),\hfill & \hfill & R_{1221}={\displaystyle \frac{1}{4}}\left(4\beta -2\alpha +{\alpha }^2\right).\hfill \end{array}$$

The Ricci curvature tensor, in matrix notation, is as follows

$$\mathrm{Ric}=\left(\begin{array}{ccc}-\frac{1}{2}{\alpha }^2& 0& 0\\ 0& \frac{1}{2}(\alpha +1)(\alpha -2\beta )& -\frac{1}{2}\alpha +\beta \\ 0& -\frac{1}{2}\alpha +\beta & -\frac{1}{2}(\alpha -1)(\alpha -2\beta )\end{array}\right).$$

Considering an LI vector field $X={\sum }_{i=1}^3{x}_i{e}_i$, then

$${\mathcal{L}}_{X}g=\left(\begin{array}{ccc}0& \frac{1}{2}\left(x_2+(2\alpha -2\beta -1)x_3\right)& \frac{1}{2}\left(x_3+(2\beta -2\alpha -1)x_2\right)\\ \frac{1}{2}\left(x_2+(2\alpha -2\beta -1)x_3\right)& -x_1& x_1\\ \frac{1}{2}\left(x_3+(2\beta -2\alpha -1)x_2\right)& x_1& -x_1\end{array}\right).$$

The scalar curvature is computed by

$$R=\frac{1}{2}{\alpha }^2-2\alpha \beta ,$$

and the contravariant tensor $\left(P^{ij}\right)$ linked to the CC is

$$P^{ij}=\left(\begin{array}{ccc}-\frac{3}{4}{\alpha }^2+\alpha \beta & 0& 0\\ 0& \frac{1}{2}\left(\frac{1}{2}{\alpha }^2-2\beta +\alpha \right)& -\beta +\frac{1}{2}\alpha \\ 0& -\beta +\frac{1}{2}\alpha & -\frac{1}{2}\left(\frac{1}{2}{\alpha }^2+2\beta -\alpha \right)\end{array}\right).$$

The inverse matrix $\left(V_{ij}\right)={\left(P^{ij}\right)}^{-1}$ can be expressed as

$$V_{ij}={\displaystyle \frac{-16}{{\alpha }^4}}\left(\begin{array}{ccc}-{\displaystyle \frac{\frac{1}{16}{\alpha }^4}{-\frac{3}{4}{\alpha }^2+\alpha \beta }}& 0& 0\\ 0& -\frac{1}{2}\left(\frac{1}{2}{\alpha }^2-\alpha +2\beta \right)& -\frac{1}{2}\alpha +\beta \\ 0& -\frac{1}{2}\alpha +\beta & \frac{1}{2}\left(\frac{1}{2}{\alpha }^2+\alpha -2\beta \right)\end{array}\right).$$

Hence, the CC tensor $\left(h_{ij}\right)$ is determined by the relation

$$h_{ij}=-\left(-\frac{3}{4}{\alpha }^2+\alpha \beta \right)V_{ij}.$$

Type III.

If the minimal polynomial of the linear transformation L possesses a root of multiplicity three, then there exists an orthonormal Lorentzian basis $\{e_1,e_2,e_3\}$ with signature $(+,+,-)$ in which the operator L admits the following matrix form:

$$L=\left(\begin{array}{ccc}\alpha & \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}& \alpha & 0\\ -\frac{1}{\sqrt{2}}& 0& \alpha \end{array}\right).$$

This induces a Lie algebra structure ${\mathfrak{g}}_{III}$ defined by the following bracket relations:

$$\left\{\begin{array}{c}[e_1,e_2]=-\frac{1}{\sqrt{2}}{e}_1-\alpha e_3,\hfill \\ [e_1,e_3]=-\frac{1}{\sqrt{2}}{e}_1-\alpha e_2,\hfill \\ [e_2,e_3]=\alpha e_1+\frac{1}{\sqrt{2}}{e}_2-\frac{1}{\sqrt{2}}{e}_3.\hfill \end{array}\right.$$

Hence, ∇ is obtained as

$$\left({\nabla }_{e_i}{e}_j\right)=\left(\begin{array}{ccc}\frac{1}{\sqrt{2}}{e}_2-\frac{1}{\sqrt{2}}{e}_3& -\frac{1}{\sqrt{2}}{e}_1-\frac{\alpha }{2}{e}_3& -\frac{1}{\sqrt{2}}{e}_1-\frac{\alpha }{2}{e}_2\\ \frac{\alpha }{2}{e}_3& \frac{1}{\sqrt{2}}{e}_3& \frac{1}{\sqrt{2}}{e}_2+\frac{\alpha }{2}{e}_1\\ \frac{\alpha }{2}{e}_2& \frac{1}{\sqrt{2}}{e}_3-\frac{\alpha }{2}{e}_1& \frac{1}{\sqrt{2}}{e}_2\end{array}\right).$$

The curvature tensor components that do not vanish are as follows:

$$\begin{array}{cc}\hfill R_{1223}& =\frac{1}{\sqrt{2}}\alpha ,R_{1231}=1,R_{2323}=\frac{1}{4}{\alpha }^2,\hfill \\ \hfill R_{1313}& =-\frac{1}{4}{\alpha }^2+1,R_{1221}=\frac{1}{4}(4+{\alpha }^2).\hfill \end{array}$$

The Ricci curvature tensor in matrix form reads

$$\mathrm{Ric}=\left(\begin{array}{ccc}-\frac{1}{2}{\alpha }^2& -\frac{1}{\sqrt{2}}\alpha & -\frac{1}{\sqrt{2}}\alpha \\ -\frac{1}{\sqrt{2}}\alpha & -\frac{1}{2}{\alpha }^2-1& -1\\ -\frac{1}{\sqrt{2}}\alpha & -1& -1+\frac{1}{2}{\alpha }^2\end{array}\right).$$

Given an arbitrary LI vector field of the form $X={\sum }_i{x}_i{e}_i$, we have

$${\mathcal{L}}_{X}g={\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{ccc}-2(x_2+x_3)& x_1& x_1\\ x_1& 2x_3& x_3-x_2\\ x_1& x_3-x_2& -2x_2\end{array}\right).$$

The scalar curvature scalar evaluates to

$$R=-\frac{3}{2}{\alpha }^2.$$

The contravariant tensor $P^{ij}$ related to the CC is

$$P^{ij}=\left(\begin{array}{ccc}\frac{1}{4}{\alpha }^2& -\frac{1}{\sqrt{2}}\alpha & \frac{1}{\sqrt{2}}\alpha \\ -\frac{1}{\sqrt{2}}\alpha & \frac{1}{4}{\alpha }^2-1& 1\\ \frac{1}{\sqrt{2}}\alpha & 1& -1-\frac{1}{4}{\alpha }^2\end{array}\right).$$

Its inverse $\left(V_{ij}\right)$ is expressed as

$$V_{ij}={\displaystyle \frac{-16}{{\alpha }^4}}\left(\begin{array}{ccc}-\frac{1}{4}{\alpha }^2& -\frac{1}{\sqrt{2}}\alpha & -\frac{1}{\sqrt{2}}\alpha \\ -\frac{1}{\sqrt{2}}\alpha & -3-\frac{1}{4}{\alpha }^2& -3\\ -\frac{1}{\sqrt{2}}\alpha & -3& -3+\frac{1}{4}{\alpha }^2\end{array}\right).$$

Therefore, the CC tensor components $\left(h_{ij}\right)$ are

$$h_{ij}=-{\displaystyle \frac{{\alpha }^2}{4}}{V}_{ij}.$$

2.2. Non-Unimodular Lie Groups

We now direct our attention to the class of non-unimodular Lie algebras. Consider a solvable Lie algebra $\mathfrak{g}$, and let us denote by $\mathfrak{G}$ the subclass where the Lie bracket $[x,y]$ always belongs to the linear span of x and y for all $x,y\in \mathfrak{g}$. Following the classification presented in [32], the non-unimodular Lorentzian Lie algebras that do not fit into $\mathfrak{G}$ and are devoid of constant sectional curvature can be realized, up to isomorphism, by an appropriate choice of basis $\{e_1,e_2,e_3\}$ with Lie brackets:

$$\left({\mathfrak{g}}_{IV}\right):[e_2,e_3]=\delta e_2+\gamma e_1,[e_1,e_3]=\beta e_2+\alpha e_1,[e_1,e_2]=0.$$

subject to the trace condition $\alpha +\delta \ne 0$, and with cases divided as follows:

IV.1 

The basis $\{e_1,e_2,e_3\}$ forms an orthonormal frame with Lorentzian metric signature determined by $⟨e_3,e_3⟩=⟨e_2,e_2⟩=-⟨e_1,e_1⟩=1$, and the structure constants satisfy $\alpha \gamma =\beta \delta$.

IV.2 

The orthonormal basis satisfies $-⟨e_3,e_3⟩=⟨e_2,e_2⟩=⟨e_1,e_1⟩=1$, along with the constraint $\beta \delta +\alpha \gamma =0$.

IV.3 

The basis $\{e_1,e_2,e_3\}$ is pseudo-orthonormal with metric tensor expressed by

$$⟨·,·⟩=\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& -1\\ 0& -1& 0\end{array}\right),$$

and the structural relation $\alpha \gamma =0$ holds.

Case IV.1.

In this setting, we get

$$\left({\nabla }_{e_i}{e}_j\right)=\left(\begin{array}{ccc}\alpha e_3& -{\displaystyle \frac{\beta -\gamma }{2}}{e}_3& \alpha e_1+{\displaystyle \frac{\beta -\gamma }{2}}{e}_2\\ -{\displaystyle \frac{\beta -\gamma }{2}}{e}_3& -\delta e_3& -{\displaystyle \frac{\beta -\gamma }{2}}{e}_1+\delta e_2\\ -{\displaystyle \frac{\beta +\gamma }{2}}{e}_2& -{\displaystyle \frac{\beta +\gamma }{2}}{e}_1& 0\end{array}\right)$$

and

$$\begin{array}{cc}\hfill R_{2323}& ={\displaystyle \frac{1}{4}}\left(2\beta \gamma +4{\delta }^2-3{\gamma }^2+{\beta }^2\right),\hfill \\ \hfill R_{1313}& ={\displaystyle \frac{1}{4}}\left(2\beta \gamma +{\gamma }^2-3{\beta }^2+4{\alpha }^2\right),\hfill \\ \hfill R_{1212}& ={\displaystyle \frac{1}{4}}\left(-2\beta \gamma +4\alpha \delta +{\gamma }^2+{\beta }^2\right).\hfill \end{array}$$

The Ricci tensor $Ric$ is diagonal with entries

$$\begin{array}{cc}\hfill R_{11}& =\frac{1}{2}\left(2(\delta \alpha +{\alpha }^2)+{\gamma }^2-{\beta }^2\right),\hfill \\ \hfill R_{22}& =-\frac{1}{2}\left(2({\delta }^2+\alpha \delta )+{\beta }^2-{\gamma }^2\right),\hfill \\ \hfill R_{33}& =-\frac{1}{2}\left(2({\delta }^2+{\alpha }^2)-{(\gamma -\beta )}^2\right).\hfill \end{array}$$

while all off-diagonal components vanish.

For an LI vector field $X=x_1{e}_1+x_2{e}_2+x_3{e}_3$, we have

$${\mathcal{L}}_{X}g=\left(\begin{array}{ccc}-2\alpha x_3& (\beta -\gamma )x_3& \alpha x_1+\gamma x_2\\ (\beta -\gamma )x_3& 2\delta x_3& -\beta x_1-\delta x_2\\ \alpha x_1+\gamma x_2& -\beta x_1-\delta x_2& 0\end{array}\right).$$

The scalar curvature R is expressed as

$$R={\displaystyle \frac{1}{2}}({\beta }^2-{\gamma }^2-4{\alpha }^2-4{\delta }^2-2\beta \gamma -4\alpha \delta ).$$

The tensor $P^{ij}$ connected to the CC tensor possesses the following nonzero diagonal entries:

$$\begin{array}{cc}\hfill P^{11}& ={\displaystyle \frac{1}{4}}\left(-2\beta \gamma -4{\delta }^2+{\gamma }^2-{\beta }^2\right),\hfill \\ \hfill P^{22}& ={\displaystyle \frac{1}{4}}\left(2\beta \gamma +3{\gamma }^2-3{\beta }^2+4{\alpha }^2\right),\hfill \\ \hfill P^{33}& ={\displaystyle \frac{1}{4}}\left(4\alpha \delta -2\beta \gamma +3{\gamma }^2+{\beta }^2\right).\hfill \end{array}$$

Hence, the CC tensor $h_{ij}$ has nonzero components given by the following:

$$\left\{\begin{array}{c}{h}_{11}=-{\displaystyle \frac{1}{16}}(4{\alpha }^2-3{\beta }^2+3{\gamma }^2+2\beta \gamma )({\beta }^2+3{\gamma }^2-2\beta \gamma +4\alpha \delta ),\hfill \\ h_{22}=-{\displaystyle \frac{1}{16}}(-{\beta }^2+{\gamma }^2-4{\delta }^2-2\beta \gamma )({\beta }^2+3{\gamma }^2-2\beta \gamma +4\alpha \delta ),\hfill \\ h_{33}=-{\displaystyle \frac{1}{16}}(-{\beta }^2+{\gamma }^2-4{\delta }^2-2\beta \gamma )(4{\alpha }^2-3{\beta }^2+3{\gamma }^2+2\beta \gamma ).\hfill \end{array}\right.$$

Case IV.2.

For Type IV.2 Lie groups, with an orthonormal basis $\{e_1,e_2,e_3\}$ where $-⟨e_3,e_3⟩=⟨e_2,e_2⟩=⟨e_1,e_1⟩=1$, ∇ is described as

$$\left({\nabla }_{e_i}{e}_j\right)=\left(\begin{array}{ccc}\alpha e_3& {\displaystyle \frac{\beta +\gamma }{2}}{e}_3& \alpha e_1+{\displaystyle \frac{\gamma +\beta }{2}}{e}_2\\ {\displaystyle \frac{\beta +\gamma }{2}}{e}_3& \delta e_3& {\displaystyle \frac{\gamma +\beta }{2}}{e}_1+\delta e_2\\ -{\displaystyle \frac{\beta -\gamma }{2}}{e}_2& {\displaystyle \frac{\beta -\gamma }{2}}{e}_1& 0\end{array}\right).$$

The curvature tensor’s non-trivial components read as follows:

$$\begin{array}{cc}\hfill R_{2323}& ={\displaystyle \frac{1}{4}}\left(-2\beta \gamma -4{\delta }^2-3{\gamma }^2+{\beta }^2\right),\hfill \\ \hfill R_{1331}& ={\displaystyle \frac{1}{4}}\left(2\beta \gamma -{\gamma }^2+3{\beta }^2+4{\alpha }^2\right),\hfill \\ \hfill R_{1212}& =\alpha \delta -{\displaystyle \frac{1}{4}}{(\gamma +\beta )}^2.\hfill \end{array}$$

The Ricci tensor is diagonal with components

$$\begin{array}{cc}\hfill R_{11}& =-{\displaystyle \frac{1}{2}}\left(-2(\alpha \delta +{\alpha }^2)-{\gamma }^2+{\beta }^2\right),\hfill \\ \hfill R_{22}& =-{\displaystyle \frac{1}{2}}\left(-2(\delta \alpha +{\delta }^2)-{\beta }^2+{\gamma }^2\right),\hfill \\ \hfill R_{33}& =-{\displaystyle \frac{1}{2}}\left(2({\delta }^2+{\alpha }^2)+{(\gamma +\beta )}^2\right).\hfill \end{array}$$

and all off-diagonal terms vanish. For an LI vector field $X=x_1{e}_1+x_2{e}_2+x_3{e}_3$, we arrive at

$${\mathcal{L}}_{X}g=\left(\begin{array}{ccc}2\alpha x_3& (\beta +\gamma )x_3& -\alpha x_1-\gamma x_2\\ (\beta +\gamma )x_3& 2\delta x_3& -\beta x_1-\delta x_2\\ -\alpha x_1-\gamma x_2& -\beta x_1-\delta x_2& 0\end{array}\right).$$

The scalar curvature is computed as

$$R={(\alpha +\delta )}^2+{\alpha }^2+{\delta }^2+{\displaystyle \frac{1}{2}}{(\beta +\gamma )}^2.$$

The symmetric tensor $P^{ij}$ involved in CC computations has diagonal entries

$$\begin{array}{cc}\hfill P^{11}& ={\displaystyle \frac{1}{4}}\left({\beta }^2-3{\gamma }^2-4{\delta }^2-2\beta \gamma \right),\hfill \\ \hfill P^{22}& ={\displaystyle \frac{1}{4}}\left(-4{\alpha }^2-3{\beta }^2+{\gamma }^2-2\beta \gamma \right),\hfill \\ \hfill P^{33}& ={\displaystyle \frac{1}{4}}\left(-{\beta }^2-{\gamma }^2-2\beta \gamma +\alpha \gamma \right).\hfill \end{array}$$

with all other entries equal to zero. Therefore, the CC tensor h has nonzero components given by

$$\left\{\begin{array}{c}{h}_{11}=-{\displaystyle \frac{1}{16}}(-4{\alpha }^2-3{\beta }^2+{\gamma }^2-2\beta \gamma )(-{\beta }^2-{\gamma }^2-2\beta \gamma +\alpha \gamma ),\hfill \\ h_{22}=-{\displaystyle \frac{1}{16}}({\beta }^2-3{\gamma }^2-4{\delta }^2-2\beta \gamma )(-{\beta }^2-{\gamma }^2-2\beta \gamma +\alpha \gamma ),\hfill \\ h_{33}=-{\displaystyle \frac{1}{16}}({\beta }^2-3{\gamma }^2-4{\delta }^2-2\beta \gamma )(-4{\alpha }^2-3{\beta }^2+{\gamma }^2-2\beta \gamma ).\hfill \end{array}\right.$$

Type IV.3.

In this setting, ∇ takes the form

$$\left({\nabla }_{e_i}{e}_j\right)=\left(\begin{array}{ccc}\alpha e_2& \frac{\gamma }{2}{e}_2& \alpha e_1-\frac{\gamma }{2}{e}_3\\ \frac{\gamma }{2}{e}_2& 0& \frac{\gamma }{2}{e}_1\\ -\beta e_2-\frac{\gamma }{2}{e}_3& -\frac{\gamma }{2}{e}_1-\delta e_2& -\beta e_1+\delta e_3\end{array}\right).$$

The curvature tensor exhibits non-vanishing components as follows:

$$\begin{array}{ccc}& & R_{1213}=\frac{1}{4}{\gamma }^2,R_{1331}={\alpha }^2-\alpha \delta +\beta \gamma ,R_{2332}=\frac{3}{4}{\gamma }^2.\hfill \end{array}$$

The corresponding Ricci tensor has the matrix representation

$$\begin{array}{c}\hfill Ric=\left(\begin{array}{ccc}-\frac{1}{2}{\gamma }^2& 0& 0\\ 0& 0& -\frac{1}{2}{\gamma }^2\\ 0& -\frac{1}{2}{\gamma }^2& -({\alpha }^2-\alpha \delta +\beta )\end{array}\right).\end{array}$$

Given a general LI vector field $X=x_1{e}_1+x_2{e}_2+x_3{e}_3$, ${\mathcal{L}}_{X}g$ is computed as

$$\left({\mathcal{L}}_{X}g\right)=\left(\begin{array}{ccc}2\alpha x_3& \gamma x_3& -\alpha x_1-\gamma x_2-\beta x_3\\ \gamma x_3& 0& -\delta x_3\\ -\alpha x_1-\gamma x_2-\beta x_3& -\delta x_3& 2(\beta x_1+\delta x_2)\end{array}\right).$$

Hence, the scalar curvature takes the value

$$R=\frac{1}{2}{\gamma }^2.$$

The matrix form of the operator $P^{ij}$ associated with the CC framework is as follows

$$\begin{array}{c}\hfill \left(P^{ij}\right)=\left(\begin{array}{ccc}-\frac{3}{4}{\gamma }^2& 0& 0\\ 0& -({\alpha }^2-\alpha \delta +\beta )& -\frac{1}{4}{\gamma }^2\\ 0& -\frac{3}{4}{\gamma }^2& 0\end{array}\right),\end{array}$$

while the associated inverse matrix $V_{ij}$ reads

$$\begin{array}{c}\hfill \left(V_{ij}\right)=\left(\begin{array}{ccc}-\frac{4}{3{\gamma }^2}& 0& 0\\ 0& 0& -\frac{4}{3{\gamma }^2}\\ 0& -\frac{4}{{\gamma }^2}& \frac{16}{3{\gamma }^4}({\alpha }^2-\alpha \delta +\beta )\end{array}\right).\end{array}$$

Consequently, the CC tensor $h_{ij}$ is obtained as

$$\begin{array}{c}\hfill \left(h_{ij}\right)=\left(\begin{array}{ccc}\frac{3}{16}{\gamma }^4& 0& 0\\ 0& 0& \frac{3}{16}{\gamma }^4\\ 0& \frac{9}{16}{\gamma }^4& -\frac{3}{4}{\gamma }^2({\alpha }^2+\beta -\alpha \delta )\end{array}\right).\end{array}$$

A manifold is locally conformally flat if, for each point in M, there exists a neighborhood and a smooth function (the conformal factor) such that the metric on the neighborhood is conformal to the flat metric. The classification of 3DLLGs that are locally conformally flat was thoroughly investigated by Calvaruso in [8]. The following reformulated statement summarizes the characterization of such manifolds:

Proposition 1.

Consider a 3DLLG $(G,g)$. The manifold $(G,g)$ is locally conformally flat if and only if it satisfies one of the following mutually exclusive conditions:

I.

$(G,g)$ is locally symmetric and complies with one of the following classifications:

(a) 

Type Ia, characterized by the equality $\alpha =\beta =\gamma$, or alternatively, any permutation thereof where $\gamma =0$ and $\alpha =\beta$;

(b) 

Type II, with both structure constants α and β identically zero;

(c) 

Type IV.1, either exhibiting constant sectional curvature, or satisfying $\alpha =\beta =\gamma =0$ with $\delta \ne 0$, or the case $\beta =\gamma =\delta =0$ while $\alpha \ne 0$;

(d) 

Type IV.2, endowed with constant curvature, or under the conditions $\alpha =\beta =\gamma =0$ and $\delta \ne 0$, or $\beta =\gamma =\delta =0$ with $\alpha \ne 0$;

(e) 

Type IV.3, either flat or satisfying $\gamma =\delta =0$ with $\alpha \ne 0$;

(f) 

Type $\mathfrak{G}$, invariably possessing constant sectional curvature.

II.

$(G,g)$ is not locally symmetric and corresponds to one of the following scenarios:

(a) 

Type Ib, with structure constants constrained by $\alpha =-2\gamma$ and $\beta =\pm \sqrt{3}\gamma$;

(b) 

Type III, where the parameter α vanishes identically;

(c) 

Type IV.3, with $\gamma =0$ and the non-degeneracy condition $\alpha \delta (\alpha -\delta )\ne 0$.

In our notation, the spatial metric components are parametrized by functions $\alpha ,\beta ,\gamma ,\delta$.

3. Generalized CCSs on 3DLLGs

This section focuses on an in-depth investigation of LI solutions to Equation (4) as posed on the 3DLLGs outlined earlier in Section 2. Through the resolution of the corresponding algebraic system, we provide a full classification of all admissible LI generalized CCSs compatible with these Lie group geometries.

Theorem 1.

Let $\mathfrak{g}$ be a unimodular 3DLLG of Type Ia. The existence of an LI generalized CCS on this structure is characterized precisely by the fulfillment of the following conditions:

• The structure constants satisfy $\alpha =\beta =\gamma$ with $\alpha \ne 0$, and the corresponding soliton constant is given by $\lambda ={\displaystyle \frac{1}{8}}{\alpha }^4-3\rho {\alpha }^2$. In this configuration, every LI vector field on $\mathfrak{g}$ is Killing.

Proof. 

To determine when a Type Ia Lie group supports a generalized CCS, we analyze the conditions imposed by Equation (4). This reduces to solving the following system:

$$\left\{\begin{array}{l}(\beta -\gamma )x_1=(\gamma -\alpha )x_2=(\alpha -\beta )x_3=0,\\ -{\displaystyle \frac{1}{8}}(2\beta \gamma -2\alpha \gamma +2\alpha \beta +{\gamma }^2-3{\beta }^2+{\alpha }^2)(-2\beta \gamma -2\alpha \gamma +2\alpha \beta +3{\gamma }^2-{\beta }^2-{\alpha }^2)\\ =\lambda -\rho (-2\beta \gamma -2\alpha \gamma +2\alpha \beta +{\gamma }^2+{\beta }^2+{\alpha }^2),\\ -{\displaystyle \frac{1}{8}}(-2\beta \gamma +2\alpha \gamma +2\alpha \beta +{\gamma }^2+{\beta }^2-3{\alpha }^2)(-2\beta \gamma -2\alpha \gamma +2\alpha \beta +3{\gamma }^2-{\beta }^2-{\alpha }^2)\\ =\lambda -\rho (-2\beta \gamma -2\alpha \gamma +2\alpha \beta +{\gamma }^2+{\beta }^2+{\alpha }^2),\\ -{\displaystyle \frac{1}{8}}(-2\beta \gamma +2\alpha \gamma +2\alpha \beta +{\gamma }^2+{\beta }^2-3{\alpha }^2)(2\beta \gamma -2\alpha \gamma +2\alpha \beta +{\gamma }^2-3{\beta }^2+{\alpha }^2)\\ =-\lambda +\rho (-2\beta \gamma -2\alpha \gamma +2\alpha \beta +{\gamma }^2+{\beta }^2+{\alpha }^2).\end{array}\right.$$

(5)

Starting with the first line, if we set $\alpha =\beta$, then either $\gamma =\alpha$ or $x_2=0$. Assuming $\gamma =\alpha$, it follows that all three structure constants are equal. Since $\left(P^{ij}\right)$ is assumed to be invertible, we must have $\alpha \ne 0$. Substituting into the remaining equations of (5), we obtain the following:

$$\lambda ={\displaystyle \frac{1}{8}}{\alpha }^4-3\rho {\alpha }^2,$$

and the soliton Equation (4) holds true for every LI vector field X.

Now, assume that $\alpha =\beta$ while $\gamma$ differs from $\alpha$. This implies $x_2=0$, and considering the relation $(\beta -\gamma )x_1=0$, it follows that $x_1$ must also vanish. The last two equations simplify to the following:

$$\left\{\begin{array}{c}-{\displaystyle \frac{1}{8}}{\gamma }^2(3{\gamma }^2-4\alpha \gamma )=\lambda -\rho ({\gamma }^2-4\alpha \gamma ),\hfill \\ -{\displaystyle \frac{1}{8}}{\gamma }^4=-\lambda +\rho ({\gamma }^2-4\alpha \gamma ).\hfill \end{array}\right.$$

Since $\gamma \ne 0$ (from invertibility of $P^{ij}$), it follows that $\gamma =\alpha$, contradicting the initial assumption.

Now consider $\alpha \ne \beta$. Then $x_3=0$, and we must analyze two subcases. First, if $\gamma =\alpha$, then the system reduces to the following:

$$\left\{\begin{array}{c}-{\displaystyle \frac{1}{8}}{\beta }^2(-3{\beta }^2+4\alpha \beta )=-\lambda +\rho ({\beta }^2-4\alpha \beta ),\hfill \\ {\displaystyle \frac{1}{8}}{\beta }^4=-\lambda +\rho ({\beta }^2-4\alpha \beta ).\hfill \end{array}\right.$$

Solving yields $\alpha =\beta$, which again contradicts our assumption.

The only remaining situation is $\alpha \ne \beta$, $\gamma \ne \alpha$, and $\beta \ne \gamma$. From the first equation, all components $x_1$, $x_2$, and $x_3$ must vanish. Substituting into the curvature equations yields the following:

$$\left\{\begin{array}{c}{\alpha }^2+\beta \gamma ={\beta }^2+\alpha \gamma ,\hfill \\ {\gamma }^2+\alpha \beta ={\beta }^2+\alpha \gamma .\hfill \end{array}\right.$$

This system has no solution under the stated inequalities, completing the proof. □

As a direct consequence of Proposition 1 and Theorem 1, we deduce the following corollary:

Corollary 1.

Any LI generalized CCS defined on a unimodular 3DLLG Type Ia is necessarily locally conformally flat.

Theorem 2.

No LI generalized CCSs exist on unimodular 3DLLGs of Type Ib.

Proof. 

Referring to Equation (4), a generalized CCS of Type Ib exists precisely when the subsequent system is satisfied:

$$\left\{\begin{array}{c}{\displaystyle \frac{1}{8}}{({\alpha }^2+4{\beta }^2)}^2+2{\beta }^2{(\alpha -2\gamma )}^2+\rho \left({\alpha }^2-4{\beta }^2-4\alpha \gamma \right)=\lambda ,\hfill \\ \beta x_2+(\alpha -\gamma )x_3=0,\hfill \\ \beta x_3+(\gamma -\alpha )x_2=0,\hfill \\ {\displaystyle \frac{1}{8}}(-3{\alpha }^2-4{\beta }^2+4\alpha \gamma )({\alpha }^2+4{\beta }^2)+2\beta x_1+\rho \left({\alpha }^2-4{\beta }^2-4\alpha \gamma \right)=\lambda ,\hfill \\ {\displaystyle \frac{1}{2}}(-3{\alpha }^2-4{\beta }^2+4\alpha \gamma )\beta (\alpha -2\gamma )=0,\hfill \\ -{\displaystyle \frac{1}{8}}(-3{\alpha }^2-4{\beta }^2+4\alpha \gamma )({\alpha }^2+4{\beta }^2)+2\beta x_1-\rho \left({\alpha }^2-4{\beta }^2-4\alpha \gamma \right)=-\lambda .\hfill \end{array}\right.$$

(6)

From the combination of the fourth and sixth relations, it follows that $4\beta x_1=0$. Given $\beta \ne 0$, one deduces $x_1=0$. The fifth condition implies either $\alpha =2\gamma$ or $-3{\alpha }^2-4{\beta }^2+4\alpha \gamma =0$.

Assuming the latter, substitution into the fourth equation yields $\lambda =-2\rho ({\alpha }^2+4{\beta }^2\gamma )$, which conflicts with the first equation unless $\beta =0$. This contradicts the assumption $\beta \ne 0$.

Alternatively, if $\alpha =2\gamma$, the first and fourth equations reduce to ${\displaystyle \frac{1}{8}}{({\alpha }^2+4{\beta }^2)}^2=\lambda +\rho ({\alpha }^2+4{\beta }^2)$ and $-{\displaystyle \frac{1}{8}}{({\alpha }^2+4{\beta }^2)}^2=\lambda +\rho ({\alpha }^2+4{\beta }^2)$, respectively. These two equalities imply $\lambda =-\rho ({\alpha }^2+4{\beta }^2)$ and $\beta =0$, again contradicting the premise $\beta \ne 0$.

Therefore, system (6) admits no solutions, and consequently, no homogeneous generalized CCSs of Type Ib exist. □

Theorem 3.

Consider the three-dimensional unimodular Lorentzian Lie algebra of Type II, denoted by ${\mathfrak{g}}_{II}$. The classification of LI generalized CCSs on ${\mathfrak{g}}_{II}$ is as follows:

• The parameters satisfy $\alpha =\beta \ne 0$, with the scalar λ expressed by $\lambda ={\displaystyle \frac{1}{8}}{\alpha }^4-3\rho {\alpha }^2$. The components fulfill $x_1=-{\displaystyle \frac{1}{4}}{\alpha }^3$ and $x_2=x_3$, valid for any real ρ.

Proof. 

The condition for a generalized CCS on Type II, as given by Equation (4), reduces to solving the following system:

$$\left\{\begin{array}{c}{\displaystyle \frac{1}{8}}{\alpha }^4+\rho ({\alpha }^2-4\alpha \beta )=\lambda ,\hfill \\ x_2+(2\alpha -2\beta -1)x_3=0,\hfill \\ x_3+(2\beta -2\alpha -1)x_2=0,\hfill \\ \left({\displaystyle \frac{1}{2}}{\alpha }^2-\alpha +2\beta \right)\left(-{\displaystyle \frac{3}{4}}{\alpha }^2+\alpha \beta \right)+x_1+\rho ({\alpha }^2-4\alpha \beta )=\lambda ,\hfill \\ (\alpha -2\beta )\left(-{\displaystyle \frac{3}{4}}{\alpha }^2+\alpha \beta \right)-x_1=0,\hfill \\ -\left({\displaystyle \frac{1}{2}}{\alpha }^2+\alpha -2\beta \right)\left(-{\displaystyle \frac{3}{4}}{\alpha }^2+\alpha \beta \right)+x_1-\rho ({\alpha }^2-4\alpha \beta )=-\lambda .\hfill \end{array}\right.$$

(7)

By aggregating the last three equations, one finds that

$$\lambda -\rho ({\alpha }^2-4\alpha \beta )={\displaystyle \frac{1}{2}}{\alpha }^2\left(-{\displaystyle \frac{3}{4}}{\alpha }^2+\alpha \beta \right).$$

Substituting this expression into the first equation of system (7) imposes $\alpha =\beta$. The invertibility of $\left(P^{ij}\right)$ guarantees that $\alpha$ cannot vanish. This subsequently forces $x_2=x_3$ and $x_1=-{\displaystyle \frac{1}{4}}{\alpha }^3$. □

Corollary 2.

For a Type II unimodular Lorentzian Lie group, possessing local conformal flatness does not automatically guarantee the existence of an LI generalized CCS.

Theorem 4.

Consider the three-dimensional unimodular Lorentzian Lie algebra ${\mathfrak{g}}_{III}$ of Type III. The associated LI generalized CCS is defined by the following conditions:

$$x_1={\displaystyle \frac{1}{2}}{\alpha }^3,x_2=-x_3=-{\displaystyle \frac{3\sqrt{2}}{4}}{\alpha }^2,\lambda ={\displaystyle \frac{1}{8}}{\alpha }^4-3\rho {\alpha }^2,\alpha \ne 0,\mathit{for}\mathit{all}\rho .$$

Proof. 

The soliton Equation (4) specialized to Type III yields the following system:

$$\left\{\begin{array}{c}{\displaystyle \frac{1}{8}}{\alpha }^4+{\displaystyle \frac{2}{\sqrt{2}}}(x_2+x_3)-3\rho {\alpha }^2=\lambda ,\hfill \\ {\displaystyle \frac{1}{2\sqrt{2}}}{\alpha }^3-{\displaystyle \frac{1}{\sqrt{2}}}{x}_1=0,\hfill \\ {\displaystyle \frac{3}{2}}{\alpha }^2+{\displaystyle \frac{1}{8}}{\alpha }^4-{\displaystyle \frac{2}{\sqrt{2}}}{x}_3-3\rho {\alpha }^2=\lambda ,\hfill \\ {\displaystyle \frac{3}{2}}{\alpha }^2-{\displaystyle \frac{1}{\sqrt{2}}}(x_3-x_2)=0,\hfill \\ {\displaystyle \frac{3}{2}}{\alpha }^2-{\displaystyle \frac{1}{8}}{\alpha }^4+{\displaystyle \frac{2}{\sqrt{2}}}{x}_2+3\rho {\alpha }^2=-\lambda .\hfill \end{array}\right.$$

(8)

From the second relation, it immediately follows that

$$x_1={\displaystyle \frac{1}{2}}{\alpha }^3.$$

Combining the first and third equations leads to

$${\displaystyle \frac{3}{2}}{\alpha }^2-{\displaystyle \frac{4}{\sqrt{2}}}{x}_3-{\displaystyle \frac{2}{\sqrt{2}}}{x}_2=0,$$

and similarly, the first and fifth imply

$${\displaystyle \frac{3}{2}}{\alpha }^2+{\displaystyle \frac{4}{\sqrt{2}}}{x}_2+{\displaystyle \frac{2}{\sqrt{2}}}{x}_3=0.$$

Solving these simultaneously results in

$$x_3=-x_2={\displaystyle \frac{3\sqrt{2}}{4}}{\alpha }^2.$$

□

Theorem 5.

In the case of a three-dimensional non-unimodular Lorentzian Lie algebra $\mathfrak{g}$ of Type IV.1, every LI generalized CCS can be categorized into one of four unique classes:

1. 

When $\alpha =\beta =0$, with $x_2=x_3=0$ and ${\gamma }^2={\delta }^2$, the scalar parameter is given by

$$\lambda ={\displaystyle \frac{9}{8}}{\gamma }^4-5\rho {\gamma }^2,$$

valid for arbitrary $x_1$, δ, ρ, and nonzero γ.

2. 

If $\alpha \ne 0$ and the relation $\gamma ={\displaystyle \frac{\beta \delta }{\alpha }}$ hold, together with $\beta =ϵ\alpha$ where ${ϵ}^2=1$, then

$$\lambda ={\displaystyle \frac{1}{8}}{\left({\alpha }^2+3{\gamma }^2+2ϵ\alpha \gamma \right)}^2-\rho \left(3{\alpha }^2+5{\gamma }^2+6ϵ\alpha \gamma \right),$$

with $x_1=-{\displaystyle \frac{\gamma }{\alpha }}{x}_2$, $x_3=0$, and arbitrary δ, ρ, $x_2$.

3. 

Under the assumptions $\alpha \ne 0$, $\delta =\gamma =x_1=x_3=0$, and ${\alpha }^2={\beta }^2$,

$$\lambda ={\displaystyle \frac{1}{8}}{\beta }^4-3\rho {\beta }^2,$$

for arbitrary $x_2$ and ρ.

4. 

If $\alpha =\delta \ne 0$ and $\beta =\gamma$ with ${\beta }^2\ne {\alpha }^2$, all components $x_1=x_2=x_3=0$ vanish. The scalar λ is

$$\lambda ={\displaystyle \frac{1}{2}}{(2{\alpha }^2+{\beta }^2)}^2-2\rho (6{\alpha }^2+{\beta }^2),$$

for arbitrary ρ.

Proof. 

Starting from the soliton Equation (4), we derive the following system:

$$\left\{\begin{array}{c}(\beta -\gamma )x_3=0,\hfill \\ \alpha x_1+\gamma x_2=0,\hfill \\ -\beta x_1-\delta x_2=0,\hfill \\ -{\displaystyle \frac{1}{8}}(4{\alpha }^2-3{\beta }^2+3{\gamma }^2+2\beta \gamma )({\beta }^2+3{\gamma }^2-2\beta \gamma +4\alpha \delta )+2\alpha x_3\hfill \\ -\rho ({\beta }^2-{\gamma }^2-4{\alpha }^2-4{\delta }^2-2\beta \gamma -4\alpha \delta )=-\lambda ,\hfill \\ -{\displaystyle \frac{1}{8}}(-{\beta }^2+{\gamma }^2-4{\delta }^2-2\beta \gamma )({\beta }^2+3{\gamma }^2-2\beta \gamma +4\alpha \delta )-2\delta x_3\hfill \\ +\rho ({\beta }^2-{\gamma }^2-4{\alpha }^2-4{\delta }^2-2\beta \gamma -4\alpha \delta )=\lambda ,\hfill \\ -{\displaystyle \frac{1}{8}}(-{\beta }^2+{\gamma }^2-4{\delta }^2-2\beta \gamma )(4{\alpha }^2-3{\beta }^2+3{\gamma }^2+2\beta \gamma )\hfill \\ +\rho (-4\alpha \delta -2\beta \gamma -4{\delta }^2-4{\alpha }^2-{\gamma }^2+{\beta }^2)=\lambda .\hfill \end{array}\right.$$

(9)

Starting with the scenario where $\alpha =0$, the invertibility of $\left(P^{ij}\right)$ imposes that $\beta =0$ while $\delta$ remains nonzero. The first equation then imposes $x_3=0$. Setting $\alpha =\beta =x_3=0$ in the final three equations yields

$$\lambda ={\displaystyle \frac{9}{8}}{\gamma }^4-\rho ({\gamma }^2+4{\delta }^2),\lambda =-{\displaystyle \frac{3}{8}}{\gamma }^2({\gamma }^2-4{\delta }^2)-\rho ({\gamma }^2+4{\delta }^2),$$

implying the identity ${\gamma }^2={\delta }^2$ and hence $\lambda ={\displaystyle \frac{9}{8}}{\gamma }^4-5\rho {\gamma }^2$. The second equation further dictates $x_2=0$. This corresponds to the first classification above.

Next, for $\alpha \ne 0$, from the second equation of (9), we find $\gamma ={\displaystyle \frac{\beta \delta }{\alpha }}$ and $x_1=-{\displaystyle \frac{\gamma }{\alpha }}{x}_2$. The third equation reduces to $({\beta }^2-{\alpha }^2)\delta x_2=0$. Assuming ${\beta }^2={\alpha }^2$, denote $\beta =ϵ\alpha$ and $\delta =ϵ\gamma$ with ${ϵ}^2=1$. The last three equations simplify, resulting in

$$\alpha x_3=0,$$

which with $\alpha \ne 0$ forces $x_3=0$. This case defines the second solution family.

If $\delta =0$, the constraint $\alpha \gamma -\beta \delta =0$ implies $\gamma =0$ and $x_1=0$. The reduced equations become

$$\left\{\begin{array}{c}-{\displaystyle \frac{1}{8}}{\beta }^2(4{\alpha }^2-3{\beta }^2)+2\alpha x_3-\rho ({\beta }^2-4{\alpha }^2)=-\lambda ,\hfill \\ {\displaystyle \frac{1}{8}}{\beta }^4+\rho ({\beta }^2-4{\alpha }^2)=\lambda ,\hfill \\ {\displaystyle \frac{1}{8}}{\beta }^2(4{\alpha }^2-3{\beta }^2)+\rho ({\beta }^2-4{\alpha }^2)=\lambda ,\hfill \end{array}\right.$$

which yield $\alpha x_3=0$ and ${\alpha }^2={\beta }^2$. Since $\alpha \ne 0$, it follows $x_3=0$. This represents the third family.

Finally, consider $\alpha \ne 0$, ${\beta }^2\ne {\alpha }^2$, $\delta \ne 0$, and $x_2=0$. From the third equation, $x_1=0$ follows. The first equation enforces either $\beta =\gamma$ or $x_3=0$. Assuming $x_3=0$, the system reduces to

$$\left\{\begin{array}{c}\begin{array}{cc}& -{\displaystyle \frac{1}{8}}\left(4{\alpha }^2-3{\beta }^2+3{\gamma }^2+2\beta \gamma \right)\left({\beta }^2+3{\gamma }^2-2\beta \gamma +4\alpha \delta \right)\hfill \\ & -\rho \left({\beta }^2-{\gamma }^2-4{\alpha }^2-4{\delta }^2-2\beta \gamma -4\alpha \delta \right)=-\lambda ,\hfill \\ & -{\displaystyle \frac{1}{8}}\left(-{\beta }^2+{\gamma }^2-4{\delta }^2-2\beta \gamma \right)\left({\beta }^2+3{\gamma }^2-2\beta \gamma +4\alpha \delta \right)\hfill \\ & +\rho \left({\beta }^2-{\gamma }^2-4{\alpha }^2-4{\delta }^2-2\beta \gamma -4\alpha \delta \right)=\lambda ,\hfill \\ & -{\displaystyle \frac{1}{8}}\left(-{\beta }^2+{\gamma }^2-4{\delta }^2-2\beta \gamma \right)\left(4{\alpha }^2-3{\beta }^2+3{\gamma }^2+2\beta \gamma \right)\hfill \\ & +\rho \left({\beta }^2-{\gamma }^2-4{\alpha }^2-4{\delta }^2-2\beta \gamma -4\alpha \delta \right)=\lambda .\hfill \end{array}\hfill \end{array}\right.$$

Given that $\left(P^{ij}\right)$ is invertible, this simplifies to

$$\left\{\begin{array}{c}{\alpha }^2+\beta \gamma ={\beta }^2+\alpha \delta ,\hfill \\ {\beta }^2+{\delta }^2={\gamma }^2+{\alpha }^2.\hfill \end{array}\right.$$

By substituting $\gamma ={\displaystyle \frac{\beta \delta }{\alpha }}$ and assuming $\alpha +\delta \ne 0$, one obtains $\alpha =\delta$, $\beta =\gamma$, and

$$\lambda ={\displaystyle \frac{1}{2}}{(2{\alpha }^2+{\beta }^2)}^2-2\rho (6{\alpha }^2+{\beta }^2),$$

which describes the fourth family.

If one attempts the alternative with $x_3\ne 0$ and $\beta =\gamma$, from the last equation it follows

$$\lambda ={\displaystyle \frac{1}{2}}(2{\delta }^2+{\beta }^2)(2{\alpha }^2+{\beta }^2)-\rho (4{\alpha }^2+4{\delta }^2+2{\beta }^2-4\alpha \delta ).$$

Substitution into the preceding equations yields

$$x_3={\displaystyle \frac{1}{2\alpha }}(2{\alpha }^2+{\beta }^2)(\alpha \delta -{\delta }^2)={\displaystyle \frac{1}{2\delta }}(2{\delta }^2+{\beta }^2)(\alpha \delta -{\alpha }^2).$$

For $x_3\ne 0$, this forces $\alpha \ne \delta$, which contradicts the assumption after simplification and leads to $\alpha =0$. Hence, this scenario is invalid. □

Corollary 3.

Local conformal flatness does not imply the existence of LI generalized CCSs on a Type IV.1 non-unimodular Lorentzian Lie group.

Theorem 6.

Let $\mathfrak{g}$ be a three-dimensional non-unimodular Lorentzian Lie algebra of Type IV.2. The full classification of LI generalized CCSs on $\mathfrak{g}$ falls into one of the following four exclusive categories:

1. 

When $\alpha =\beta =0$, with $x_2=x_3=0$, and parameters satisfying $\delta \ne 0$, ${\delta }^2={\gamma }^2$, the scalar parameter satisfies

$$\lambda ={\displaystyle \frac{1}{8}}{\gamma }^4+5\rho {\gamma }^2,$$

while $x_1$ remains arbitrary.

2. 

For $\alpha =\delta \ne 0$ and $\gamma =-\beta =4\alpha$, all $x_i$ vanish, and

$$\lambda =2{\alpha }^4+12\rho {\alpha }^2,$$

valid for any ρ.

3. 

If $\alpha =\delta \ne 0$, $\gamma =-\beta$, with $x_1=x_2=0$, then

$$\lambda =2{\beta }^4+12\rho {\alpha }^2,x_3=-{\displaystyle \frac{1}{4}}{\alpha }^2\beta -{\alpha }^3,$$

for arbitrary ρ.

4. 

When $\alpha \ne 0$, $\delta \ne 0$, $\gamma =-\beta =-4\delta$, and $x_1=x_2=0$, the scalar and vector components satisfy

$$\lambda =2{\beta }^2{\delta }^2+2\rho \left({(\alpha +\delta )}^2+{\alpha }^2+{\delta }^2\right),x_3=-{\alpha }^2\delta -\alpha {\delta }^2,$$

for any ρ.

Proof. 

Starting from the soliton Equation (4), the system reduces to the following:

$$\left\{\begin{array}{c}(\beta +\gamma )x_3=0,\hfill \\ -\alpha x_1-\gamma x_2=0,\hfill \\ -\beta x_1-\delta x_2=0,\hfill \\ -{\displaystyle \frac{1}{8}}(-4{\alpha }^2-3{\beta }^2+{\gamma }^2-2\beta \gamma )(-{\beta }^2-{\gamma }^2-2\beta \gamma +\alpha \gamma )-2\alpha x_3\hfill \\ +2\rho \left({(\alpha +\delta )}^2+{\alpha }^2+{\delta }^2+\frac{1}{2}{(\beta +\gamma )}^2\right)=\lambda ,\hfill \\ -{\displaystyle \frac{1}{8}}({\beta }^2-3{\gamma }^2-4{\delta }^2-2\beta \gamma )(-{\beta }^2-{\gamma }^2-2\beta \gamma +\alpha \gamma )-2\delta x_3\hfill \\ +2\rho \left({(\alpha +\delta )}^2+{\alpha }^2+{\delta }^2+\frac{1}{2}{(\beta +\gamma )}^2\right)=\lambda ,\hfill \\ -{\displaystyle \frac{1}{8}}({\beta }^2-3{\gamma }^2-4{\delta }^2-2\beta \gamma )(-4{\alpha }^2-3{\beta }^2+{\gamma }^2-2\beta \gamma )\hfill \\ -2\rho \left({(\alpha +\delta )}^2+{\alpha }^2+{\delta }^2+\frac{1}{2}{(\beta +\gamma )}^2\right)=-\lambda .\hfill \end{array}\right.$$

Case 1: Suppose $\alpha =0$. Invertibility of $\left(P^{ij}\right)$ demands $\beta =0$ and $\delta \ne 0$. The first equation yields $x_3=0$. Substituting $\alpha =\beta =x_3=0$ into the last three equations results in

$$\lambda ={\displaystyle \frac{1}{8}}{\gamma }^4+2\rho \left(2{\delta }^2+\frac{1}{2}{\gamma }^2\right),\lambda =-{\displaystyle \frac{1}{8}}{\gamma }^2(3{\gamma }^2+4{\delta }^2)+2\rho \left(2{\delta }^2+\frac{1}{2}{\gamma }^2\right).$$

Consistency forces ${\gamma }^2={\delta }^2$. From the second equation, $x_2=0$. This characterizes the first solution family.

Case 2: For $\alpha \ne 0$, the constraint $\alpha \gamma +\beta \delta =0$ gives $\gamma =-{\displaystyle \frac{\beta \delta }{\alpha }}$. The second and third equations imply $x_1=-{\displaystyle \frac{\gamma }{\alpha }}{x}_2$ and $\delta x_2=0$.

If $\delta =0$, then $\gamma =0$ and $x_1=0$. The system reduces to

$$\left\{\begin{array}{c}-{\displaystyle \frac{1}{8}}{\beta }^2(4{\alpha }^2+3{\beta }^2)+2\alpha x_3-2\rho \left(2{\alpha }^2+\frac{1}{2}{\beta }^2\right)=-\lambda ,\hfill \\ {\displaystyle \frac{1}{8}}{\beta }^4+2\rho \left(2{\alpha }^2+\frac{1}{2}{\beta }^2\right)=\lambda ,\hfill \\ {\displaystyle \frac{1}{8}}{\beta }^2(4{\alpha }^2+3{\beta }^2)+2\rho \left(2{\alpha }^2+\frac{1}{2}{\beta }^2\right)=\lambda ,\hfill \end{array}\right.$$

implying $4{\alpha }^2+2{\beta }^2=0$, contradicting $\alpha \ne 0$.

Case 3: If $\alpha \ne 0$, $\delta \ne 0$, and $x_2=0$, then $x_1=0$. The first equation yields either $\beta =-\gamma$ or $x_3=0$.

Assuming $x_3=0$, the last three equations reduce to a linear system that forces

$$\left\{\begin{array}{c}{\beta }^2+{\alpha }^2={\gamma }^2+{\delta }^2,\hfill \\ {\beta }^2+{\alpha }^2+\beta \gamma ={\displaystyle \frac{1}{4}}\alpha \gamma .\hfill \end{array}\right.$$

Replacing $\gamma =-{\displaystyle \frac{\beta \delta }{\alpha }}$ and assuming $\alpha +\delta \ne 0$ yield $\alpha =\delta$, $\beta =-\gamma$, and $\gamma =4\alpha$. Then,

$$\lambda =2{\alpha }^4+12\rho {\alpha }^2,$$

identifying the second family.

Case 4: Suppose $\alpha \ne 0$, $\delta \ne 0$, $x_2=0$, $x_3\ne 0$, and $\beta =-\gamma$. Then $x_1=0$, and the sixth equation determines

$$\lambda =2{\alpha }^2{\delta }^2+12\rho {\alpha }^2.$$

Inserting this into the fourth and fifth equations yields

$$x_3=-{\displaystyle \frac{1}{4}}{\alpha }^2\beta -\alpha {\delta }^2,x_3=-{\displaystyle \frac{1}{4}}\alpha \beta \delta -{\alpha }^2\delta .$$

Equating the two expressions for $x_3$ leads to

$$(-\delta +\frac{1}{4}\beta )(-\delta +\alpha )=0.$$

If $\alpha =\delta$, then

$$x_3=-{\displaystyle \frac{1}{4}}{\alpha }^2\beta -{\alpha }^3,$$

defining the third family. Otherwise, if $\alpha \ne \delta$, then $\beta =4\delta$ and

$$x_3=-{\alpha }^2\delta -\alpha {\delta }^2,$$

which corresponds to the fourth family. □

Corollary 4.

A locally conformally flat Type IV.2 non-unimodular Lorentzian Lie group may fail to possess any LI generalized CCS.

Theorem 7.

There exist no LI generalized CCSs on any three-dimensional non-unimodular Lorentzian Lie group of Type IV.3.

Proof. 

The soliton Equation (4) for Type IV.3 reduces to solving the following:

$$\left\{\begin{array}{c}{\displaystyle \frac{3}{8}}{\gamma }^4-2\alpha x_3+\rho {\gamma }^2=\lambda ,\hfill \\ \gamma x_3=0,\hfill \\ -\alpha x_1-\gamma x_2-\beta x_3=0,\hfill \\ \lambda =\rho {\gamma }^2,\hfill \\ -\alpha x_1-\gamma x_2-\beta x_3+2\delta x_3-\rho {\gamma }^2=-\lambda ,\hfill \\ -{\displaystyle \frac{3}{2}}{\gamma }^2({\alpha }^2-\alpha \delta +\beta )-2(\beta x_1+\delta x_2)=0.\hfill \end{array}\right.$$

Since $\left(P^{ij}\right)$ is invertible, we have $\gamma \ne 0$. Then $\gamma x_3=0$ implies $x_3=0$. The relation $\alpha \gamma =0$ enforces $\alpha =0$. Substituting $\alpha =0$ into the first and fourth equations yields $\gamma =0$, contradicting $\gamma \ne 0$. Therefore, no such soliton exists. □

4. Conclusions

In this paper, we classified left-invariant generalized cross-curvature solitons on simply connected three-dimensional Lorentzian Lie groups, under the standing assumption that the contravariant tensor $P_{ij}$ is invertible. For each Lie algebra type considered (Ia, Ib, II, III, IV.1, IV.2, IV.3), we obtained explicit algebraic conditions on the structure constants that are necessary and sufficient for the existence of a left-invariant generalized cross-curvature soliton, described the admissible left-invariant vector fields X, and computed the soliton parameter $\lambda$ as a function of the structure constants and parameter $\rho$. Our main findings include existence families for Types Ia (locally conformally flat), II, and III and various families in IV.1 and IV.2; nonexistence results for Type Ib and IV.3; and multiple examples illustrating how the parameter $\rho$ affects the algebraic conditions.

Limitations of the present work include the assumption of left-invariance and the invertibility of $P_{ij}$; degenerate cases where $P_{ij}$ fails to be invertible were listed and left for future study. Natural directions for further research are (i) the analysis of degenerate (non-invertible) cases and their modified soliton equations; (ii) dynamical stability of the homogeneous solitons under generalized cross-curvature flow; and (iii) extensions to other signatures or to higher dimensions where appropriate analogues exist.