Polynomial Conserved Quantities of Lie Applicable Surfaces

Using the gauge theoretic approach for Lie applicable surfaces, we characterise certain subclasses of surfaces in terms of polynomial conserved quantities. These include isothermic and Guichard surfaces of conformal geometry and $L$-isothermic surfaces of Laguerre geometry. In this setting one can see that the well known transformations available for these surfaces are induced by the transformations of the underlying Lie applicable surfaces. We also consider linear Weingarten surfaces in this setting and develop a new B\"{a}cklund-type transformation for these surfaces.


Introduction
In [15,16,14], Demoulin defined a class of surfaces satisfying the equation given in terms of curvature line coordinates (u, v), where U is a function of u, V is a function of v, ǫ ∈ {0, 1, i}, E and G denote the usual coefficients of the first fundamental form and κ 1 and κ 2 denote the principal curvatures. In the case that ǫ = 0, one calls these surfaces Ω-surfaces and if ǫ = 0 we call them Ω 0 -surfaces. Together, Ω-and Ω 0 -surfaces form the applicable surfaces of Lie sphere geometry (see [3]). By using the hexaspherical coordinate model of Lie [27] it is shown in [31] that these surfaces are the deformable surfaces of Lie sphere geometry. This gives rise to a gauge theoretic approach for these surfaces which is developed in [13]. That is, the definition of Lie applicable surfaces is equated to the existence of a certain 1-parameter family of flat connections. This approach lends itself well to the transformation theory of these surfaces. The gauge theoretic approach is explored further in [34], for which this paper is meant as a sequel. In [11,37] a gauge theoretic approach for isothermic surfaces in Möbius geometry is developed. By considering polynomial conserved quantities of the arising 1-parameter family of flat connections, one can characterise familiar subclasses of surfaces in certain space forms. For example, constant mean curvature surfaces in space forms are characterised by the existence of linear conserved quantities [4,11]. By then applying the transformation theory of the underlying isothermic surface, one obtains transformations for these subclasses. In this paper we apply this framework to Lie applicable surfaces. For example, we show that isothermic surfaces, Guichard surfaces and L-isothermic surfaces are Ω-surfaces admitting a linear conserved quantity. This is particularly beneficial to the study of the transformations of these surfaces. For example, we will show that the Eisenhart transformation for Guichard surfaces (see [17]), which was given a conformally invariant treatment in [4], is induced by the Darboux transformation of the underlying Ω-surface. One can also show (see [33]) that the special Ω-surfaces of [19] can be characterised as Ω-surfaces admitting quadratic conserved quantities, however we will not explore this further in this paper.
In [8,9] linear Weingarten surfaces in space forms are characterised as Lie applicable surfaces whose isothermic sphere congruences take values in certain sphere complexes. In this paper we shall review this theory from the viewpoint of polynomial conserved quantities. We shall see that non-tubular linear Weingarten surfaces in space forms are Ω-surfaces admitting a 2-dimensional vector space of linear conserved quantities, whereas tubular linear Weingarten surfaces are Ω 0 -surfaces admitting a constant conserved quantity. By using this approach we obtain a new Bäcklund-type transformation for linear Weingarten surfaces.
Acknowledgements. We would like to thank G. Szewieczek for reading through this paper and providing many useful comments. This work has been partially supported by the Austrian Science Fund (FWF) through the research project P28427-N35 "Non-rigidity and Symmetry breaking" as well as by FWF and the Japan Society for the Promotion of Science (JSPS) through the FWF/JSPS Joint Project grant I1671-N26 "Transformations and Singularities". The fourth author was also supported by the two JSPS grants Grant-in-Aid for Scientific Research (C) 15K04845 and (S) 24224001 (PI: M.-H. Saito).

Preliminaries
Given a vector space V and a manifold Σ, we shall denote by V the trivial bundle Σ × V . Given a vector subbundle W of V , we define the derived bundle of W , denoted W (1) , to be the subset of V consisting of the images of sections of W and derivatives of sections of W with respect to the trivial connection on V . In this paper, most of the derived bundles that appear will be vector subbundles of the trivial bundle, but in general this is not always the case as, for example, the rank of the derived bundle may not be constant over Σ.
Throughout this paper we shall be considering the pseudo-Euclidean space R 4,2 , i.e., a 6-dimensional vector space equipped with a non-degenerate symmetric bilinear form ( , ) of signature (4,2). Let L denote the lightcone of R 4,2 . The orthogonal group O(4, 2) acts transitively on L. The lie algebra o(4, 2) of O (4,2) is well known to be isomorphic to the exterior algebra ∧ 2 R 4,2 via the identification for a, b, c ∈ R 4,2 . We shall frequently use this fact throughout this paper.
Proof. If q is timelike then q ⊥ has signature (4, 1) and cannot contain the 2dimensional lightlike subspace f (p) for each p ∈ Σ.
Suppose that q is spacelike and that on some open subset U ⊂ Σ, f ≤ q ⊥ . Without loss of generality, assume that U = Σ. Then this implies that f (1) ≤ q ⊥ and q ∈ Γf ⊥ . Hence, f (1) = f ⊥ , contradicting Remark 2.2.
Therefore, if f ≤ q ⊥ then the only possibility left to consider is that q is lightlike. Then since the maximal lightlike subspaces of R 4,2 are 2-dimensional, q ∈ Γf ⊥ if and only if q ∈ Γf . By Lemma 2.4 this is the case only if f is totally umbilic.
We shall often refer to a non-zero vector q ∈ R 4,2 as a sphere complex. As Lemma 2.5 shows, for a Legendre map f : Σ → Z, generically f ∩ q ⊥ defines a rank 1 subbundle of f .

Conformal geometry.
Let p ∈ R 4,2 such that p is not lightlike. If p is timelike then p ⊥ ∼ = R 4,1 and defines a Riemannian conformal geometry. If p is spacelike then p ⊥ ∼ = R 3,2 and defines a Lorentzian conformal geometry. We consider elements of P(L ∩ p ⊥ ) to be points and refer to p as a point sphere complex.
The elements of P(L\ p ⊥ ) give rise to spheres in the following way: suppose that s ∈ P(L\ p ⊥ ). Now s ⊕ p is a (1, 1)-plane and thus V := (s ⊕ p ) ⊥ is a (3, 1)-plane. The projective lightcone of V is then diffeomorphic to S 2 and we thus identify V with a sphere in P(L ∩ p ⊥ ).
Conversely, suppose that V ≤ p ⊥ is a (3, 1)-plane. Then V ⊥ is a (1, 1)-plane in R 4,2 containing p and we identify the two null lines of V ⊥ with the sphere defined by V with opposite orientations.
Remark 2.7. Those Lie sphere transformations that fix the point sphere complex are the conformal transformations of p ⊥ .
Suppose that f : Σ → Z is a Legendre map. Then, by Lemma 2.5, on a dense open subset of Σ, Λ := f ∩ p ⊥ is a rank 1 subbundle of f . Using the identification of ∧ 2 R 4,2 with the skew-symmetric endomorphisms on R 4,2 , we have for any τ ∈ Γ(∧ 2 f ) that τ p ∈ Γf and, since τ is skew-symmetric, τ p ⊥ p. Hence, Away from points where Λ ⊥ q, we have that for any nowhere zero τ ∈ Γ(∧ 2 f ), are the projections of f into Q 3 and P 3 , respectively. We can then write f = f, t .
Definition 2.8. We call f the space form projection of f and t the tangent plane 1 congruence of f .
One can easily see that: Lemma 2.9. The space form projection of f into Q 3 exists at p ∈ Σ if and only if the kernel of the linear map is trivial at p.
Away from umbilic points, suppose that (u, v) are curvature line coordinates for f. Then by Rodrigues' equations we have that where κ 1 and κ 2 are the principal curvatures of f. Therefore, s 1 := t + κ 1 f and s 2 := t + κ 2 f are curvature spheres of f with respective curvature subbundles T 1 := ∂ ∂u and T 2 := ∂ ∂v . 2.2.2. Laguerre geometry. In this subsection we shall recall the correspondence given in [12] between Lie sphere geometry and Laguerre geometry. Let q ∞ ∈ L and define U := P(L)\ q ∞ ⊥ . Then (E, ψ) with E := {y ∈ L : (y, q ∞ ) = −1} and ψ : E → U, y → [y] defines an affine chart for U . Choosing q 0 ∈ L such that (q 0 , q ∞ ) = −1, we have that q 0 , q ∞ ⊥ ∼ = R 3,1 . We may then define the orthogonal projection Then π • ψ −1 defines an isomorphism between U and q 0 , q ∞ ⊥ . We thus identify points in U with points in R 3,1 . Now let W := P(L ∩ q ∞ ⊥ )\ q ∞ . Then π identifies W with the projective lightcone of q 0 , q ∞ ⊥ and thus P(L 3 ), where L 3 is the lightcone of R 3,1 . Therefore, we identify W with null directions in R 3,1 . We define q ∞ to be the improper point of Laguerre geometry.
Under this correspondence, contact elements in R 4,2 are then identified with affine null lines in R 3,1 , i.e., for z ∈ R 3,1 and l ∈ P(L 3 ) By choosing a point sphere complex p ∈ q 0 , q ∞ ⊥ with |p| 2 = −1, we have that One identifies points in R 3,1 with oriented spheres (including point spheres, but not oriented planes) in R 3 in the following way: a sphere centred at x ∈ R 3 with signed radius r ∈ R is identified with the point This is classically known as isotropy projection [3,12]. We then have that null lines in R 3,1 correspond to pencils of spheres in R 3 in oriented contact with each other and isotropic planes in R 3,1 are identified with oriented planes in R 3 . It was shown in [12] that the Lie sphere transformations A ∈ O(4, 2) that preserve the improper point q ∞ are identified under this correspondence with the affine Laguerre transformations of R 3,1 , that is, the identity component of the group R 4 ⋊ O(3, 1). In terms of transformations of R 3 , this group consists of the Lie sphere transformations that map oriented planes to oriented planes.
Defining Q 3 := {y ∈ L : (y, q ∞ ) = −1, (y, p) = 0}, we have that π| Q 3 is an isometry between Q 3 and q 0 , q ∞ , p ⊥ and this restricts to the usual Euclidean projection in the conformal geometry defined by p ⊥ , see [11,37,23] . We say that f is a Lie applicable surface if there exists a closed η ∈ Ω 1 (f ∧ f ⊥ ) such that [η ∧ η] = 0 and the quadratic differential q defined by is non-zero. Furthermore, if q is non-degenerate (respectively, degenerate) on a dense open subset of Σ we say that f is an Ω-surface (Ω 0 -surface).
Given a closed η ∈ Ω 1 (f ∧ f ⊥ ), we have for any τ ∈ Γ(∧ 2 f ) thatη := η − dτ is a new closed 1-form taking values in Ω 1 (f ∧ f ⊥ ). We then say thatη and η are gauge equivalent and this yields an equivalence relation on closed 1-forms with values in f ∧ f ⊥ . We call the equivalence class the gauge orbit of η. As shown in [34,Corollary 3.3], q is well defined on gauge orbits, i.e., if η andη are gauge equivalent then q =q, for their respective quadratic differentials.
Let us assume that f is umbilic-free. Then there are two distinct curvature sphere congruences s 1 and s 2 with respective curvature subbundles T 1 and T 2 .  One has a splitting of the trivial bundle called the Lie cyclide splitting: Therefore,we may split a closed 1-form η into η = η h + η m , accordingly. In [34,Definition 3.8] it is shown that there is a unique member of the gauge orbit of η that satisfies η m ∈ Ω 1 (∧ 2 f ). We call this unique member the middle potential and denote it by η mid .
Assumption: for the rest of this paper we will make the assumption that the signature of the quadratic differential q is constant over all of Σ.
From Proposition 2.11, one can deduce that q ∈ Γ((T * 1 ) 2 ⊕ (T * 2 ) 2 ). Therefore, after possibly rescaling q by ±1 and switching T 1 and T 2 , we may write for unique (up to sign) lifts of the curvature sphere congruences σ 1 ∈ Γs 1 and σ 2 ∈ Γs 2 . The middle potential is then given by where ⋆ is the Hodge-star operator of the conformal structure c for which the curvature directions on T Σ are null. One finds that q is divergence-free with respect to c, i.e., in terms of curvature line coordinates u and v, there exist functions U of u and V of v such that When one projects to a space form, where the space form projection immerses, one finds that Demoulin's equation is satisfied. By gauging η mid by ±ǫσ 1 ∧ σ 2 , we obtain closed 1-forms Thus, s ± := σ 1 ± ǫσ 2 are isothermic sphere congruences (see [6,23]). There then exist (unique up to constant reciprocal rescaling) lifts σ ± ∈ Γs ± such that We call these lifts the Christoffel dual lifts of s ± . In terms of these lifts the middle potential has the form:

2.4.
Transformations of Lie applicable surfaces. The transformation theory for Lie applicable surfaces was developed in [13] and was further explored in [34]. In this section we shall review some of this theory. The richness of the transformation theory of Lie applicable surfaces follows from the following result:  In the case that we are using the middle potential, η mid , we shall refer to the 1parameter family of connections {d + tη mid } t∈R as the middle pencil of connections, or for brevity, the middle pencil.

Calapso transforms.
For each t ∈ R and gauge potential η, since d+tη is a flat metric connection, there exists a local orthogonal trivialising gauge transformation T (t) : Σ → O(4, 2), that is, Since (∧ 2 f )f = 0, we have that the Calapso transforms are well defined on the gauge orbit.
In [34,Theorem 4.4] it is shown that η t := Ad T (t) · η is a closed 1-form taking values in f t ∧ (f t ) ⊥ . Furthermore, [η t ∧ η t ] = 0 and q t = q. Thus we have the following theorem: Theorem 2.15. Calapso transforms are Lie applicable surfaces.
In fact, this 1-parameter family of Lie applicable surfaces arises because Lie applicable surfaces are the deformable surfaces of Lie sphere geometry (see [31]).
are the local trivialising orthogonal gauge transformations of d + sη t .

2.4.2.
Darboux transforms. Fix m ∈ R × and let η be any gauge potential. Since d m := d + mη is a flat connection, it has many parallel sections. Suppose thatŝ is a null rank 1 parallel subbundle of d m such thatŝ is nowhere orthogonal to the curvature sphere congruences of f . Let s 0 :=ŝ ⊥ ∩ f and letf := s 0 ⊕ŝ.  Recall from [11,37,6,13] that for L,L ∈ P(L) such that L ⊥L and t ∈ R × we have an orthogonal transformation In the case that f andf are umbilic-free we have the following result regarding the middle pencils of the two surfaces: whereŝ ≤f and s ≤ f are the parallel subbundles of d + mη mid and d + mη mid , respectively, implementing these Darboux transforms.
In particular, this proposition shows that given any subbundleŝ ≤f , one may choose a gauge potential η such thatŝ is a parallel subbundle of d + mη.
A pertinent question is "how many Darboux transforms does a Lie applicable surface admit?" By using that for every m ∈ R, one deduces the following lemma: Lemma 2.21.ŝ is a null rank 1 parallel subbundle of d + mη if and only ifŝ = T −1 (m)L for some constantL ∈ P(L). Now, since P(L) is 4-dimensional, d + mη admits a 4-parameter family of null rank 1 parallel subbundles. Since this holds for every m ∈ R, we obtain the following answer to our question:  has sectional curvature κ = 0 and Q 3 ∼ = R 3 . Then we may choose a null vector q 0 ∈ p ⊥ such that (q 0 , q ∞ ) = −1. Thus q ∞ , p, q 0 ⊥ ∼ = R 3 and we have an isometry We can use this to identify f := f ∩ Q 3 with a surface x : Σ → R 3 . Let n : Σ → S 2 denote the unit normal of x. We then have that df = dx + (dx, x)q ∞ and the tangent plane congruence of f is given by t = n + (n, x)q ∞ + p.
It was shown in [34, Section 5] that there exists a 1-parameter family of closed 1-forms η in the gauge orbit of η mid satisfying (ηp, q ∞ ) = 0. We may then write where x D andx are Combescure transforms of x, i.e., x D andx have parallel curvature directions to x, such that the principal curvatures of the surfaces satisfy This shows that {x, x D ,x, n} forms a system of O-surfaces, see [26]. Conversely, given such a system of surfaces satisfying (6), one can check that η defined in (5) is a closed 1-form, and thus f is an Ω-surface.
We call x D an associate surface of x andx an associate Gauss map of x. In [34,Theorem 5.4] it was shown that an associate surface of an Ω-surface is itself an Ω-surface.

Polynomial conserved quantities
Suppose that f : Σ → Z is a Lie applicable surface with family of flat connections {d t = d + tη} t∈R . We now give a definition that is analogous to that of [11,37]: The following lemma shows that the existence of polynomial conserved quantities is gauge invariant. Suppose thatη is in the gauge orbit of η so thatη = η − dτ for τ ∈ Γ(∧ 2 f ). From Lemma 2.13 we immediately get the following result: Using an identical argument to [11,Proposition 2.2], one obtains the following lemma: Lemma 3.3. Suppose that p is a polynomial conserved quantity of d t . Then the real polynomial (p(t), p(t)) has constant coefficients.
From now on we shall assume that f is an umbilic-free Ω-surface and assume that η is the middle potential η mid .
Since p d ∈ Γf , τ p d = 0 for any τ ∈ Γ(∧ 2 f ). Therefore, is a polynomial of degree at most d and the coefficient of Remark 3.5. For polynomial conserved quantities of Ω 0 -surfaces, 2 and 3 of Proposition 3.4 do not necessarily hold. We shall not consider general polynomial conserved quantities of Ω 0 -surfaces, however in Subsection 6.1.2 we shall consider constant conserved quantities.
Corollary 3.6. Suppose that p is a polynomial conserved quantity of degree d of the middle pencil of f . Then for τ ∈ Γ(∧ 2 f ),p(t) := exp(tτ )p(t) has degree strictly less Proof. From 3 of Proposition 3.4, we have that the coefficient of t d ofp is given by We may write τ = βσ + ∧ σ − , where β is a smooth function and σ ± are Christoffel dual lifts. Then by 2 of Proposition 3.4, we have that and (σ − , p d−1 ) both vanish as this would imply that p has degree strictly less than d. Therefore, without loss of generality, assume that ( Corollary 3.7. Suppose that p is a polynomial conserved quantity of d + tη mid . Then the degree d of p is invariant under gauge transformation if and only if (p(t), p(t)) is a polynomial of degree 2d − 1.
Proof. By 2 of Proposition 3.4 we have that p d ∈ Γf . Therefore there is no 2d-term of (p(t), p(t)). Now the coefficient of t 2d−1 in (p(t), p(t)) is 2(p d , p d−1 ) and by 2 of Therefore, by Corollary 3.6, the coefficient of t 2d−1 vanishes if and only if there exists τ ∈ Γ(∧ 2 f ) such that exp(tτ )p(t) has degree strictly less than d.
Analogously to [11,37], we make the following definition: Definition 3.8. An umbilic-free Ω-surface is a special Ω-surface of type d if the middle pencil of f admits a non-zero polynomial conserved quantity of degree d.
One should note that a special Ω-surface of type d is automatically a special Ω-surface of type d + n, for all n ∈ N, because one may always multiply p(t) by a real valued polynomial of degree n, for example, t n . On the other hand a special Ω-surface of type d can also be a special Ω-surface of lower type.
Note also that type zero special Ω-surfaces do not exist as this would imply that there exists q ∈ (R 4,2 ) × such that q ∈ Γf , implying that f is totally umbilic.
Now suppose that f is a special Ω-surface of type d with degree d conserved quantity p. Let m be a non-zero root of the polynomial (p(t), p(t)). Then p(m) is lightlike and is a parallel section of d + mη mid . If we let s 0 := f ∩ p(m) ⊥ and definef := s 0 ⊕ p(m) , thenf is a Darboux transform of f with parameter m. Unsurprisingly, [11,37] lead us to make the following definition: Definition 3.9. The Darboux transformsf of f such that p(m) ∈ Γf for some m ∈ R × are called the complementary surfaces of f with respect to p.
Since the degree of (p(t), p(t)) is less than or equal to 2d − 1, we have at most 2d − 1 complementary surfaces.

Transformations of polynomial conserved quantities
We would now like to investigate how polynomial conserved quantities behave when we apply the transformations of Subsection 2.4. Suppose that f is a special Ω-surface of type d and let p be the associated degree d polynomial conserved quantity of the middle pencil of f .
denotes the local trivialising orthogonal gauge transformations of d + tη mid . We now have a result analogous to [11,Theorem 3.12]: Proof. By Proposition 2.16, the middle pencil of f t is given by Then it follows immediately that p t is a polynomial conserved quantity of d + s(η t ) mid . Furthermore, the coefficient of We have thus proved the following Theorem: The Calapso transforms of special Ω-surfaces of type d are special Ω-surfaces of type d.

Darboux transformations.
Suppose thatf and f are umbilic-free Darboux transforms of each other with parameter m ∈ R × . Then by Proposition 2.19, the middle pencil off is given by whereŝ ≤f and s ≤ f are the parallel subbundles of d + mη mid and d + mη mid , respectively, implementing these Darboux transforms. Therefore, Γŝ is a conserved quantity of d + tη mid . Using the splitting we shall write p(t) as We then have the following proposition: Proof. First note that by Proposition 3.4, the top term p d of p(t) lies in f . Therefore, [p(t)]ŝ has degree strictly less than d. Hence, Finally, we have that in either casep(0) = p(0) because Γŝ s (1) is the identity. Since whereσ ∈ Γŝ is a parallel section of d m . Therefore, if p(m) ∈ Γŝ ⊥ at a point p ∈ Σ, then p(m) ∈ Γŝ ⊥ throughout Σ. By Lemma 2.21, one then deduces that there is a 3-parameter family of Darboux transforms with parameter m satisfying p(m) ∈ Γŝ ⊥ . Since this holds for every m ∈ R, we have the following theorem: Theorem 4.5. Darboux transforms of special Ω-surfaces of type d are special Ωsurfaces of type d + 1. Furthermore, there is a 4-parameter family of these Darboux transforms that are special Ω-surfaces of type d.

Type 1 special-Ω surfaces
In this section we shall see that special Ω-surfaces of type 1, i.e., Ω-surfaces whose middle pencil admits a linear conserved quantity p(t), include isothermic surfaces, Guichard surfaces and L-isothermic surfaces. Furthermore the familiar transformations of these surfaces are restrictions of the transformations of Subsection 2.4. For example the Eisenhart transformations for Guichard surfaces are Darboux transformations preserving the linear conserved quantity.
Suppose that f is a special Ω-surface of type 1 and let p(t) = p 0 + tp 1 be the associated linear conserved quantity of the middle pencil of f . By Proposition 3.4, p 0 is constant and p 1 ∈ Γf . We may also deduce the following lemma: Proof. The necessity of this lemma follows immediately from part 2 of Proposition 3.4. One can quickly deduce the sufficiency by using the form of the middle potential given in (4).
Let f : Σ → Z be the Legendre lift of Λ. Then Λ = f ∩ p ⊥ and η takes values in f ∧ f ⊥ . Furthermore, the quadratic differential coincides with the holomorphic 3 (with respect to the conformal structure induced by Λ) quadratic differential defined in [11,37]. Thus, q is non-degenerate and f is an Ω-surface. Furthermore, (d + tη)p = 0, i.e., p is a constant conserved quantity of d + tη. Thus, if τ ∈ Γ(∧ 2 f ) such that the middle pencil of f is given by d + tη mid = exp(tτ ) · (d + tη), then we have that p(t) = exp(tτ )p is a linear conserved quantity of d + tη mid . Moreover, (p(t), p(t)) = (p, p) is a non-zero constant.
Conversely, suppose that f is a special Ω-surface of type 1 with linear conserved quantity p and suppose that (p(t), p(t)) is a non-zero constant. If we let p := p(0), then p is a point sphere complex and p ⊥ defines a (Riemannian or Lorentzian) conformal geometry. By Corollaries 3.6 and 3.7, we have that one of the isothermic sphere congruences, without loss of generality Λ := s + , of f takes values in p ⊥ . Then Λ is an isothermic surface and is its associated closed 1-form. We have therefore arrived at the following theorem: Special Ω-surfaces of type 1 whose degree 1 polynomial conserved quantity p satisfies (p(t), p(t)) being a non-zero constant are the isothermic surfaces of the conformal geometry defined by p(0) ⊥ .
We shall now see how the classical transformations of isothermic surfaces are induced by the transformations of Subsection 2.4: suppose that f is an umbilic-free Ω-surface such that Λ := s + is an isothermic surface in p ⊥ . Then p(t) := exp(tτ )p is a polynomial conserved quantity of the middle pencil, where τ = 1 2 σ + ∧ σ − for Christoffel dual lifts σ ± . 5.1.1. Calapso transforms. Suppose that f t = T (t)f is a Calapso transform of f . Since the Calapso transforms are well defined on gauge orbits (see, Section 2.4.1), we may assume that T (t) is the gauge transformation of d + tη + . Now Thus, T (t)p is constant and, by premultiplying by an appropriate Lie sphere transformation, we may assume that it is p. Then T (t)Λ ≤ f t is a Calapso transform of the isothermic surface Λ in the sense of [11,23,37]. Thus,ŝ + := σ + is a Darboux transform in the sense of [23,37,11,6] of the isothermic surface s + .
Conversely, ifΛ is a Darboux transform of Λ with parameter m thenŝ := exp(−mτ )Λ is a parallel subbundle of d + mη mid andŝ ≤ p(m) ⊥ , since p(m) = exp(−mτ )p. We have therefore arrived at the following theorem: for some A ∈ ΓEnd(T Σ) and (η + p, q ∞ ) = 0. By comparing this with Subsection 2.5, we have that there is an associate surface x D of x such that 1 One can then deduce that the conformal structures induced by x and x D are equivalent. Therefore, since x and x D have parallel curvature directions and induce the same conformal structure, they are Christoffel transforms of each other.

Guichard surfaces.
In this subsection we shall characterise Guichard surfaces in conformal geometries amongst special Ω-surfaces of type 1. We will then see how the well known transformations of these surfaces are induced by the transformations of the underlying Ω-surface. This exposition has been partly outlined in [13,Section 7.5.2].
Suppose that p ∈ R 4,2 is a point sphere complex for a conformal geometry p ⊥ . Let L p denote the lightcone of p ⊥ . Recall from Section 2.2.1 that spheres in this conformal geometry are represented by (3, 1)-planes V ≤ p ⊥ . Therefore, given a two-dimensional manifold Σ, one can represent a sphere congruence as a rank 4 subbundle V of the bundle Σ × p ⊥ with induced signature (3, 1). One may then split the trivial connection d on R 4,2 = V ⊕ V ⊥ as where D V is the sum of the induced connections on V and V ⊥ and N V ∈ Ω 1 (V ∧ V ⊥ ). Let s ands denote the null rank 1 subbundles of V ⊥ . We may then write where N s ∈ Ω 1 (s ∧ V ) and Ns ∈ Ω 1 (s ∧ V ). We say that a map Λ : Σ → P(L p ) envelops V if Λ (1) ⊂ V , equivalently, N V Λ = 0. One then has that f := Λ ⊕ s and f := Λ ⊕s are the Legendre maps enveloping Λ with opposite orientations.
In [4], Möbius flat submanifolds are derived and studied. In codimension 1, these coincide with Guichard surfaces: Definition 5.6 ([4]). Λ is a Guichard surface if, for some (and in fact, any) enveloped sphere congruence V , there exists χ V ∈ Ω 1 (Λ ∧ Λ (1) ) such that Associated to a Guichard surface is a quadratic differential q Λ ∈ ΓS 2 (T Σ) * defined by . It is shown in [4] that this quadratic differential is independent of the choice of V . Furthermore, if q Λ is a degenerate quadratic differential, then Λ is a channel surface.
So now let us suppose that Λ is a Guichard surface with non-degenerate q Λ , i.e., Λ is a non-channel Guichard surface. Let V be an enveloped sphere congruence of Λ. Consider now where η := 2(χ V + Ns) ∈ Ω 1 (f ∧ f ⊥ ). Since d V t is flat, one has that d + t 2 −1 2 η is flat and thus η is closed. Furthermore, one deduces that the quadratic differential q of η, i.e., q(X, Y ) = tr(σ → η X d Y σ), coincides with q Λ . Thus, q is non-degenerate and f is an Ω-surface. Now p is a constant conserved quantity of d V t , thus tΓ s s (t)p is a conserved quantity of d+ t 2 −1 2 η. Moreover, one may write tΓ s s (t)p = p + t 2 −1 2 σ for some σ ∈ Γs. By reparameterising, one has that p(t) := p + tσ is a linear conserved quantity of d + tη. Furthermore, (p(t), p(t)) is a linear polynomial with non-zero constant term.

One then deduces that
Hence, Λ is Möbius flat. Moreover, q Λ coincides with q. Thus, we have proved the following theorem: Special Ω-surfaces of type 1 whose degree 1 polynomial conserved quantity p satisfies (p(t), p(t)) being linear with non-zero constant term are the nonchannel Guichard surfaces in the conformal geometry of p(0) ⊥ .

5.2.1.
Calapso transforms. Let f t := T (t)f be a Calapso transform of f . Then by Proposition 4.1, the middle pencil of f t admits a linear conserved quantity p t defined by p t (s) = T (t)p(t + s). Now since T (t) take values in O(4, 2), we have that (p t (s), p t (s)) = (p(t + s), p(t + s)).
Therefore, (p t (s), p t (s)) is a linear polynomial with constant term (p(t), p(t)). Since (p(t), p(t)) is a linear polynomial in t with non-zero constant term, it admits a single root which we shall denote t 0 . By applying Theorem 5.7, we obtain the following theorem: Theorem 5.8. If t = t 0 , then the Calapso transform f t projects to a Guichard surface in the conformal geometry of T (t)p(t) ⊥ .
Remark 5.9. The Calapso transform f t0 admits a linear polynomial p t0 such that (p t0 (s), p t0 (s)) is a linear polynomial with vanishing constant term. Therefore Theorem 5.7 does not apply in this case.
In [4] a spectral deformation is defined for Guichard surfaces Λ. Given an enveloping sphere congruence V , the 1-parameter family of connections d V t is flat for all t. Thus there exist trivialising gauge transformations Φ V t such that Φ V t · d V t = d. For r ∈ R we then say that Λ r := Φ V r Λ is a T-transform of Λ. On the other hand, we have that Hence the Calapso transforms of the underlying Ω-surface coincide with the T-transforms of Λ.

The Eisenhart transformation.
In [17], Eisenhart determines a Bäcklund type transformation for Guichard surfaces, which has come to be known as the Eisenhart transformation. A conformally invariant description of this transformation is given in [4], and we will now show how this is induced by certain Darboux transforms of the underlying Ω-surfaces.
Firstly let us assume that f is a special Ω-surface whose linear conserved quantity p of the middle pencil satisfies (p(t), p(t)) being a linear polynomial with non-zero constant term. Let Λ := f ∩ p(0) ⊥ . Then by Theorem 5.7, Λ is a Guichard surface.
Suppose thatΛ is an Eisenhart transform of Λ with parameter m. Then, by [4], there exists a sphere congruence V r enveloping Λ andΛ such thatΛ is a parallel subbundle of d Vr m . Now for the appropriate choice of gauge potential η, one has that (12) d Vr t = Γ s s (t −1 ) · (d + t 2 −1 2 η), where s ands are the rank 1 null subbundles of V ⊥ r . SinceΛ is a parallel subbundle d Vr m , one has that Γ s s (m −1 )Λ is a parallel subbundle of d+ m 2 −1 2 η. Moreover, sinceΛ is enveloped by V r ,Λ ≤ V r and thus Γ s s (m −1 )Λ =Λ. Now by definingf := s 0 ⊕Λ, where s 0 := f ∩Λ ⊥ , we have thatf is a Darboux transform of f with parameter wherep is the linear conserved quantity of η. Thus,Λ ⊥p( m 2 −1 2 ). Now when we let s ≤f denote the parallel subbundle of d + m 2 −1 2 η mid , this condition is equivalent toŝ ⊥ p( m 2 −1 2 ). Conversely, suppose thatf is a Darboux transform of f with parameter m 2 −1 2 such that the parallel subbundleŝ ≤f of d + m 2 −1 2 η mid satisfiesŝ ⊥ p( m 2 −1 2 ). Then by Proposition 2.20 there exists a gauge potential η in the gauge orbit of η mid for whichΛ =f ∩ p(0) ⊥ is a parallel subbundle. Letp be the corresponding linear conserved quantity of d + tη. Then one has thatΛ ⊥p( m 2 −1 2 ). Let V := p(0),p(1) ⊥ . Then V envelopes Λ andΛ. Furthermore,Λ is a parallel subbundle , and s ands are the null rank 1 subbundles of V ⊥ . Therefore, by [4],Λ is an Eisenhart transform of Λ with parameter m. We have therefore arrived at the following theorem: Choose a null space form vector q ∞ ∈ p ⊥ . Then Q 3 is isometric to Euclidean 3-space. As usual, let f : Σ → Q 3 denote the space form projection of f into Q 3 and let t : Σ → P 3 denote its tangent plane congruence. Now we may choose a 1-formη in the gauge orbit of η mid such that the linear conserved quantityp ofη satisfiesp 1 ⊥ q ∞ . After rescaling by a constant if necessary, one can deduce thatη has the formη for some A ∈ ΓEnd(T Σ). Therefore, by comparing with Subsection 2.5, any projection x : Σ → R 3 of f with unit normal n : Σ → S 2 admits an associate surface x D such that the associate Gauss map is given by the unit normal of x, i.e.,x = n. Thus Hence, x D is an associate surface in the sense of Guichard [22]. 5.3. L-isothermic surfaces. L-isothermic surfaces were originally discovered by Blaschke [3] and have been the subject of interest recently in for example [20,28,29,30,32,36,39]. They are the surfaces in R 3 that admit curvature line coordinates that are conformal with respect to the third fundamental form of the surface, or as Musso and Nicolodi [30] put it, there exists a holomorphic 4 (with respect to the third fundamental form) quadratic differential q that commutes with the second fundamental form, i.e., if we use the complex structure induced on Σ by III to split the second fundamental form into bidegrees, then II 2,0 = µq 2,0 , for some real valued function µ : Σ → R. In [30], L-isothermic surfaces were also characterised in terms of the standard model for Laguerre geometry R 3,1 (see for example [3,12]). In this subsection we will show that Legendre lifts of L-isothermic surfaces are the special Ω-surfaces of type 1 whose linear conserved quantity p of the middle pencil satisfies (p(t), p(t)) = 0.
Recall from Subsection 2.2.2 that a non-zero lightlike vector q ∞ defines a Laguerre geometry and that by choosing q 0 ∈ L such that (q 0 , q ∞ ) = −1 and p ∈ q 0 , q ∞ ⊥ such that |p| 2 = −1, one can show that Now suppose that f : Σ → Z is a Legendre map and that f projects to a surface f : Σ → Q 3 with tangent plane congruence t : Σ → P 3 . Then where x : Σ → R 3 is the corresponding surface in R 3 with unit normal n : Σ → S 2 . Suppose that there exists a holomorphic (with respect to the third fundamental form of x, III = (dn, dn)) quadratic differential q that commutes with the second fundamental form of x, II = −(dx, dn). This implies that if we let Q ∈ ΓEnd(T Σ) such that q = (dn, dn • Q), then Q is trace-free and symmetric with respect to III and the 2-tensor is symmetric. Now let Then dη is equal to It follows from the fact that Q is trace-free that dn dn • Q = 0. Furthermore, one can check that q being holomorphic implies that dn • Q is closed. Finally, for any X, Y ∈ ΓT Σ, is non-degenerate and ηq ∞ = 0. Hence, f is an Ω-surface and for some τ ∈ Γ(∧ 2 f ), p(t) := exp(tτ )q ∞ is a linear conserved quantity of the middle pencil satisfying Conversely, suppose that f is a special Ω-surface of type 1 whose linear conserved quantity p of the middle pencil satisfies (p(t), p(t)) = 0. Let q ∞ := p 0 . Then q ∞ is a space form vector for a space form with vanishing sectional curvature. Furthermore, by Corollaries 3.6 and 3.7, one of the isothermic sphere congruences, without loss of generality s + , takes values in q ∞ ⊥ . Let p ∈ q ∞ ⊥ be a point sphere complex with |p| 2 = −1 and let t ∈ Γs + be the lift of s + such that (t, p) = −1. Then t defines a tangent plane congruence for the space form projection f : Σ → Q 3 of f . Now η + has the form for some Q ∈ ΓEnd(T Σ). Therefore, and q is holomorphic with respect to the conformal structure induced by t. Furthermore, since η + is closed, we have that Thus, q commutes with the second fundamental form of f. Hence, f projects to an L-isothermic surface. We therefore have the following theorem: Theorem 5.11. Special Ω-surfaces of type 1 whose linear polynomial conserved quantity p satisfies (p(t), p(t)) = 0 are the L-isothermic surfaces of any Laguerre geometry defined by p(0).

5.3.1.
Calapso transforms. L-isothermic surfaces are well known to be the deformable surfaces of Laguerre geometry [28] and this gives rise to T -transforms for these surfaces [32]. Therefore it is unsurprising that the Calapso transforms of Legendre lifts of L-isothermic surfaces yield L-isothermic surfaces. Fix t ∈ R and let f t be a Calapso transform of f . Then by Proposition 4.1, the middle pencil of f t admits a linear conserved quantity p t defined by p t (s) = T (t)p(t + s).
Since T (t) takes values in O(4, 2) we have that (p t (s), p t (s)) = (p(t + s), p(t + s)) = 0. Now T (t)p(t) is a constant null vector. Therefore, by premultiplying by an appropriate Lie sphere transformation, we may assume that it is q ∞ . By applying Theorem 5.11 we obtain the following theorem: Theorem 5.12. The Calapso transforms of L-isothermic surfaces are L-isothermic.

The Bianchi-Darboux transform.
Suppose thatf is a Darboux transform of f with parameter m and suppose thatŝ ≤f is the parallel subbundle of d + mη mid . Then by Proposition 4.3, if p(m) ∈ Γŝ ⊥ , thenp(t) := Γŝ s (1 − t/m)p(t) is a linear conserved quantity of d + mη mid with (p(t),p(t)) = (p(t), p(t)) andp(0) = p(0). Hence, by Theorem 5.11,f projects to a L-isothermic surface in any space form with point sphere complex p(0). It was shown (via a lengthy computation) in [33] that this transformation coincides with the Bianchi-Darboux transformation (see for example [20,30]): Theorem 5.13. The Bianchi-Darboux transforms of an L-isothermic surface constitute a 4-parameter family of Darboux transforms of its Legendre lifts.

Associate surface.
We shall now recover the result of [38, Section 6] that L-isothermic surfaces are the Combescure transforms of minimal surfaces.
Let x : Σ → R 3 be an L-isothermic surface. Given that for some Q ∈ ΓEnd(T Σ), we have that (η + q ∞ , p) = 0. By comparing with Section 2.5, we have that there is an associate Gauss mapx of x satisfying Thus, there exists a minimal surfacex with the same spherical representation as x.
In fact, we have a converse to this result: Theorem 5.14. Suppose thatx : Σ → R 3 is a minimal surface. Then any Combescure transform x : Σ → R 3 ofx is an L-isothermic surface.
Proof. Let x be a Combescure transformation ofx, i.e., x andx have the same spherical representation. Let n be the common normal of these surfaces. Then the result follows by the fact that is a closed 1-form.
The characterisation of L-isothermic surfaces as the Combescure transforms of minimal surfaces shows that the class of L-isothermic surfaces is preserved by Combescure transformation.

Further work.
There is one case that we have not considered in this section -when f admits a linear conserved quantity p such that (p(t), p(t)) is a linear polynomial with vanishing constant term. It would be interesting to know if these surfaces have a classical interpretation in the Laguerre geometry defined by p(0). One interesting fact about these surfaces is that if we further project into a Euclidean subgeometry of p(0) ⊥ then the resulting surface is an associate surface of itself. Furthermore, by Remark 5.9 these surfaces appear as one of the Calapso transforms of a Guichard surface.

Complementary surfaces.
Suppose that f is a special Ω-surface of type 1 with linear conserved quantity p(t) = p 0 + tp 1 . Now the polynomial (p(t), p(t)) has degree less than or equal to 1 and admits non-zero roots if and only if either • (p(t), p(t)) is linear with non-zero constant term, in which case f projects to a Guichard surface in p(0) ⊥ , by Theorem 5.7, or • (p(t), p(t)) is the zero polynomial, in which case f projects to an L-isothermic surface in the Laguerre geometry defined by p(0), by Theorem 5.11. Now suppose that m is a root of p(m) and letf be the corresponding complementary surface. Now by Theorem 3.4, p 1 ∈ Γf and thus Conversely, suppose thatf is a Darboux transform of f with parameter m such that there exists a constant vector q ∈ Γ(f +f ). Letσ ∈ Γf be a parallel section of d + mη mid . Nowσ = λ q + σ for some non-zero smooth function λ and σ ∈ Γf . Thus, 0 = (d + mη mid )σ = dλ q + dσ + mλη mid q.
Therefore, p admits non-zero roots andf is complementary surface of f with respect to p.
If (p(0), p(0)) is non-zero then Hence, f andf project to the same Guichard surface in the conformal geometry p(0) ⊥ . If (p(0), p(0)) = 0 then, by Corollary 5.2, p(0) lies nowhere in f and we must have that p(0) ∈ Γf . Thus,f is totally umbilic. We have thus arrived at the following theorem: Proposition 5.15. Suppose thatf is a Darboux transform of f . Then there exists a constant vector q ∈ Γ(f +f ) if and only if f is a type 1 special Ω-surface that admitsf as a complementary surface. Furthermore, if q is lightlike then f projects to an L-isothermic surface in the Laguerre geometry defined by q andf is totally umbilic. Otherwise, f andf project to the same Guichard surface in the conformal geometry q ⊥ .
In particular, Proposition 5.15 gives us a characterisation of L-isothermic surfaces in terms of their Darboux transforms: An Ω-surface projects to an L-isothermic surface in some Laguerre geometry if and only if it admits a totally umbilic Darboux transform.

Linear Weingarten surfaces
Let f : Σ → Q 3 be the space form projection of a Legendre map f : Σ → Z into the (Riemannian or Lorentzian) space form Q 3 with constant sectional curvature κ. Let ε be +1 in the case that Q 3 is a Riemannian space form and −1 in the case that it is Lorentzian. Recall the following definition: Definition 6.1. Where f immerses we say that it is a linear Weingarten surface if (13) aK + 2bH + c = 0 for some a, b, c ∈ R, not all zero, where K = κ 1 κ 2 is the extrinsic Gauss curvature of f and H = 1 2 (κ 1 + κ 2 ) is the mean curvature of f. A special case of linear Weingarten surfaces is given by flat fronts: Definition 6.2. A surface f : Σ → Q 3 is a flat front if, where it immerses, the intrinsic Gauss curvature K int := ε K + κ vanishes.
In [8] it was shown that flat fronts in hyperbolic space are those Ω-surface whose isothermic sphere congruences each envelop a fixed sphere. In [9] it was shown that linear Weingarten surfaces in space forms correspond to Lie applicable surfaces whose isothermic sphere congruences take values in certain linear sphere complexes. This theory was discretised in [7]. In this section we shall review this theory in terms of linear conserved quantities of the middle pencil.
Recall from Section 2.2 how we break symmetry from Lie sphere geometry to space form geometry. Let q, p ∈ R 4,2 be a space form vector and point sphere complex for a space form Let f : Σ → Z be a Legendre map and assume that f projects into Q 3 . Let f : Σ → Q 3 denote the space form projection of f and let t : Σ → P 3 denote its tangent plane congruence.
Similarly to [7], we have an alternative characterisation of the linear Weingarten condition: From Proposition 6.3 one quickly deduces the observation of [9], that if f projects to a linear Weingarten surface in a space form with space form vector q and point sphere complex p then f projects to a linear Weingarten surface in any other space form with space form vector and point sphere complex chosen from q, p . 6.1. Linear Weingarten surfaces in Lie geometry. We shall now recover the results of [9] regarding the Lie applicability of umbilic-free linear Weingarten surfaces.
Proposition 6.4. f is an umbilic-free linear Weingarten surface satisfying (13) if and only if f is a Lie applicable surface with middle potential Furthermore, tubular linear Weingarten surfaces give rise to Ω 0 -surfaces and nontubular linear Weingarten surfaces give rise to Ω-surfaces whose isothermic sphere congruences are real in the case that b 2 − ac > 0 and complex conjugate in the case that b 2 − ac < 0.
Thus η is closed if and only if f is a linear Weingarten surface satisfying (13). Furthermore, one can check that, modulo Ω 1 (∧ 2 f ), η is equal to . Since t + κ 1 f ∈ Γs 1 and t + κ 2 f ∈ Γs 2 , we have that the Ω 1 (S 1 ∧ S 2 ) part of η lies in Ω 1 (∧ 2 f ). Thus η is the middle potential η mid . Now the quadratic differential induced by η mid is given by Since f is an umbilic-free immersion, i.e., κ 1 = κ 2 , q is non-zero. Moreover, Clearly, real linear combinations of polynomial conserved quantities are polynomial conserved quantities. However, the degree of the polynomials may not be preserved. For example, one can check that there exists a constant conserved quantity within the span of the conserved quantities p and q of Corollary 6.5 if and only if f is a tubular linear Weingarten surface, i.e., b 2 − ac = 0. Therefore, in the non-tubular case, any linear combination of p and q yields a linear conserved quantity of d + tη mid . In light of this we will consider 2 dimensional vector spaces of linear conserved quantities for Ω-surfaces: 6.1.1. Non-tubular linear Weingarten surfaces. Suppose that f is an Ω-surface and suppose that P is a 2 dimensional vector space of linear conserved quantities of d + tη mid . By P (t) we shall denote the subset of R 4,2 formed by evaluating P at t. Lemma 6.6. For each t ∈ R, P (t) is a rank 2 subbundle of R 4,2 .
Proof. Let p, q ∈ P . Then by Lemma 5.1, for some q 0 , p 0 ∈ R 4,2 . Then p(t) and q(t) are linearly dependent sections of P (t) for some t ∈ R if and only if p 0 and q 0 are linearly dependent if and only if p and q are linearly dependent.
We may equip P with a pencil of metrics {g t } t∈R∪{∞} defined for each t ∈ R and α, β ∈ P by g t (α, β) := (α(t), β(t)), and g ∞ := lim t→∞ 1 t g t . Thus, if we write α(t) = α 0 + tα 1 and β(t) = β 0 + tβ 1 then Then, for general t ∈ R, we have that We shall now consider the 3-dimensional vector space S 2 P formed by the abstract symmetric product on P . For each t ∈ R we can identify elements of S 2 P with symmetric endomorphisms on R 4,2 via the map Furthermore, we have an isomorphism from S 2 P to the space of symmetric tensors on P with respect to g ∞ , denoted S 2 ∞ P defined by φ ∞ : where for γ, δ ∈ P , γ)). Using Corollary 6.5, we obtain the following proposition: Proposition 6.7. Suppose that f is a non-tubular linear Weingarten surface satisfying (13). Then f is an Ω-surface whose middle pencil admits a 2-dimensional space of linear conserved quantities P with g 0 = 0 and non-degenerate g ∞ . Furthermore, the linear Weingarten condition [W ] is given by . Proof. By Proposition 6.4, f is an Ω-surface and by Corollary 6.5, P := p, q is a 2-dimensional space of linear conserved quantities for d + tη mid , where p(t) := p + t(−bf + at) and q(t) := q + t(cf − bt).
Since p is a point sphere complex, i.e., |p| 2 = 0, we have that g 0 = 0. We also have that Therefore g ∞ is non-degenerate and Remark 6.8. It follows from the proof of Proposition 6.7 that if b 2 −ac > 0 then g ∞ is indefinite and if b 2 −ac < 0 then g ∞ is definite. Then it follows by Proposition 6.4 that the isothermic sphere congruences are real when g ∞ is indefinite and complex conjugate when g ∞ is definite.
We now seek a converse to Proposition 6.7. Firstly we have the following technical lemma that gives conditions for our Ω-surface to project to a well-defined map in certain space forms, i.e., so that our point sphere map does not have points at infinity: Lemma 6.9. Suppose that q, p ∈ P (0) are a space form vector and point sphere complex for a space form Q 3 . Then f defines a point sphere map f : Σ → Q 3 with tangent plane congruence t : Σ → P 3 if and only if g ∞ is non-degenerate.
One can then deduce that By Corollary 5.2, f lies nowhere in q ⊥ or p ⊥ . It then follows by Lemma 2.9 that f defines a point sphere map f and tangent plane congruence t if and only if g ∞ is non-degenerate.
We are now in a position to state the following proposition: Proposition 6.10. Suppose that f is an umbilic-free Ω-surface whose middle pencil admits a 2-dimensional space of linear conserved quantities P , such that g 0 = 0 and g ∞ is non-degenerate. Then f projects to a non-tubular linear Weingarten surface with , where it immerses, in any space form determined by space form vector and point sphere complex q, p ∈ P (0).
If g ∞ is non-degenerate on P then g ∞ induces two null directions on P . In the case that g ∞ is indefinite these are real directions and in the case that g ∞ is definite they are complex conjugate. Let q ± be two linearly independent vectors in P ⊗ C and define q ± := q ± (0) ∈ R 4,2 ⊗ C. Then . Therefore, q ± are null with respect to ( , ) ∞ if and only if we have (after possibly switching q ± ) that (σ ± , q ± ) = 0, i.e., the isothermic sphere congruences s ± take values in q ± ⊥ . Now by applying Proposition 6.7 and Proposition 6.10 we obtain the main result of [9]: Theorem 6.11. Non-tubular linear Weingarten surfaces in space forms are those Ω-surfaces whose isothermic sphere congruences each take values in a linear sphere complex.
Furthermore, by scaling q ± appropriately we have that g ∞ = (q + ⊙q − ) ∞ . Therefore, we have that [W ] = [q + ⊙ q − ], which was shown in [7] for the discrete case. 6.1.2. Tubular linear Weingarten surfaces. In [9], the following theorem is proved: Theorem 6.12. Tubular linear Weingarten surfaces in space forms are those Ω 0surfaces whose isothermic curvature sphere congruence takes values in a linear sphere complex.
We shall recover this result in terms of our setup. Suppose that f is a tubular linear Weingarten surface satisfying (13), i.e., b 2 − ac = 0. Then by Proposition 6.4, f is an Ω 0 -surface and, by Corollary 6.5, the middle pencil of f admits conserved quantities p(t) := p + t(−bf + at) and q(t) := q + t(cf − bt).
q is a non-zero constant conserved quantity of d+tη mid . This implies that η mid q 0 = 0. Without loss of generality, assume that the middle potential has the form Then 0 = η mid q 0 = (σ 1 , q 0 ) ⋆ dσ 1 − (⋆dσ 1 , q 0 )σ 1 . Since f is umbilic-free we have that d 2 σ 1 does not take values in f and thus (σ 1 , q 0 ) = 0, i.e., s 1 ≤ q 0 ⊥ . Conversely, suppose that f is an umbilic-free Legendre map such that s 1 ≤ q 0 ⊥ . Letq 0 ∈ R 4,2 such that the plane q 0 ,q 0 is not totally degenerate. Then let [W ] ∈ P(S 2 R 4,2 ) be defined by Remark 6.14. Notice in the converse argument to Theorem 6.12 that we did not have to assume that f was an Ω 0 -surface. We can thus deduce that if one of the curvature sphere congruences of a Legendre map takes values in a linear sphere complex then it must be isothermic.

6.2.
Transformations of linear Weingarten surfaces. Using the identification of non-tubular linear Weingarten surfaces as certain Ω-surfaces, we will apply the transformations of Subsection 2.4 to obtain new linear Weingarten surfaces. Let f be an Ω-surface whose middle pencil d + tη mid admits a 2-dimensional space of linear conserved quantities P , such that g 0 = 0 and g ∞ is non-degenerate. Then, by Proposition 6.10, f projects to linear Weingarten surfaces with linear Weingarten condition , in any space form determined by space form vector and point sphere complex chosen from P (0). 6.2.1. Calapso transformations. In [9], the Calapso transformation for Ω-surfaces was used to obtain a Lawson correspondence for linear Weingarten surfaces. This was further investigated in [7] in the discrete setting. We shall recover this analysis in terms of linear conserved quantities of the middle pencil.
Let t ∈ R and consider the Calapso transform f t = T (t)f of f . For each p ∈ P we have by Proposition 4.1 that p t defined by p t (s) = T (t)p(t + s) is a linear conserved quantity of the middle pencil of f t . Therefore, the middle pencil of f t admits a 2-dimensional space of linear conserved quantities P t defined by the isomorphism Ψ : P → P t , p → p t .
As with P , we may equip P t with a pencil of metrics {g t s } s∈R∪{∞} . Then for each s ∈ R and α t , β t ∈ P , g t s (α t , β t ) = (T (t)α(t + s), T (t)β(t + s)) = (α(t + s), β(t + s)) = g t+s (α, β), by the orthogonality of T (t). Thus, Ψ is an isometry from (P, g t+s ) to (P t , g t s ). It is then clear that Ψ is an isometry from (P, g ∞ ) to (P t , g t ∞ ). Therefore, g t ∞ is non-degenerate, and g t 0 = 0 if and only if g t = 0.
Proposition 6.15. There exists t ∈ R × such that g t = 0 if and only if f projects to a flat front in any space form determined by P (0).
Proof. Since g t = g 0 + tg ∞ , for each t ∈ R × , we have that g t = 0 if and only if g 0 = −tg ∞ . Now let q, p ∈ P be an orthogonal basis with respect to g ∞ . Then [W ] is given by Thus, if g 0 = −tg ∞ then q(0) and p(0) are orthogonal and define a space form vector and point sphere complex for a space form with sectional curvature κ = −g 0 (q, q) and assuming that p is normalised such that g 0 (p, p) = ±1, ε = −g 0 (p, p). Furthermore, by Proposition 6.3, f projects to a surface f with constant extrinsic Gauss curvature K = − g ∞ (q, q) g ∞ (p, p) = − g 0 (q, q) g 0 (p, p) = − κ ε in this space form, i.e., f is a flat front.
Conversely, suppose f projects to a flat front f in a space form defined by space form vector q and point sphere complex p, i.e., f satisfies εK + κ = 0.
Since κ = −|q| 2 and ε = −|p| 2 , by Corollary 6.5 we have that p(t) = p + t |p| 2 t and q(t) = q + t |q| 2 f are linear conserved quantities of the middle pencil. Moreover, g 1 2 (p, p) = g 1 2 (q, p) = g 1 2 (q, q) = 0. Hence, g 1 2 = 0. Now consider the maps φ t s : S 2 P t → S 2 P t (s), α t ⊙ β t → α t (s) ⊙ β t (s). Then, by extending the action of Ψ to S 2 P and T (t) to S 2 R 4,2 in the standard way, one has that φ t s = T (t) • φ t+s • Ψ −1 . Furthermore, if we define φ t ∞ : S 2 P t → S 2 ∞ P t analogously to φ ∞ , then as g t ∞ is isometric to g ∞ via Ψ, we have that (φ t ∞ ) −1 g t ∞ = Ψ • φ −1 ∞ g ∞ . Applying Proposition 6.10, we have proved the following proposition: Proposition 6.16. Suppose that g t = 0. Then f t projects to a linear Weingarten surface with linear Weingarten condition , in any space form determined by space form vector and point sphere complex chosen from P t (0) = T (t)P (t).
In a similar way to [8], we have the following result regarding Calapso transforms of flat fronts: Corollary 6.17. Suppose that f projects to a flat front and let t ∈ R such that g t = 0. Then f t projects to a flat front in any space form determined by space form vector and point sphere complex chosen from P t (0) = T (t)P (t).
Proof. Recall that for any s ∈ R, g t s is isometric to g t+s via Ψ. Now by Proposition 6.15, there exists t 0 ∈ R × such that g t0 = 0. Therefore, g t t0−t = 0 and it follows by Proposition 6.15 that f t projects to a flat front in P t (0).
To summarise this section we have the following theorem: Theorem 6.18. Calapso transforms give rise to a Lawson correspondence for nontubular linear Weingarten surfaces. Furthermore,p(0) = p(0) and (p(t),p(t)) = (p(t), p(t)). Now, if we assume that P (m) ≤ŝ ⊥ , thenP is a 2-dimensional space of linear conserved quantities of the middle pencil off , where we defineP via the isomorphism Υ : P →P , p →p.
As for f , defineφ t : S 2P → S 2P (t) andφ ∞ : S 2P → S 2 ∞P accordingly. Then for each t ∈ R,φ Then the linear Weingarten condition forf is given by . Therefore, we have proved the following proposition: Proposition 6.19.f is a linear Weingarten surface with the same linear Weingarten condition as f in any space form determined by space form vector and point sphere complex chosen from P (0). By Lemma 2.21 one deduces that, for each m ∈ R × , there exists a 2-parameter family of Darboux transforms with parameter m such that P (m) ≤ s ⊥ . Therefore: Theorem 6.20. A linear Weingarten surface possesses a 3-parameter family of Darboux transforms that satisfy the same linear Weingarten condition as the initial surface.
In [2, §273] and [1, §398], Bianchi constructs a 3-parameter family of Ribaucour transformations of pseudospherical (K = −1) surfaces in Euclidean space into surfaces of the same kind by performing two successive complex conjugate Bäcklund transformations. This has been investigated recently in [21]. This transformation preserves I + III. On the other hand Proposition 6.4 tells us that I + III coincides with the quadratic differentials of the underlying Ω-surfaces. By applying Proposition 4.6, one deduces that Bianchi's family of transformations is included in the 3-parameter family of Darboux transforms detailed in Theorem 6.20.
A similar transformation exists for spherical (K = 1) surfaces in Euclidean space. It was shown in [24] and subsequently [25] that these transformations are induced by the Darboux transformations of their parallel CMC-surfaces. Recalling from Theorem 5.5 that Darboux transforms of isothermic surfaces are Darboux transforms of their Legendre lifts, one deduces that these transformations are also included in the family detailed in Theorem 6.20.