Table Of ContentOn completeness of the Ribaucour transformations for
triply orthogonal curvilinear coordinate systems in R3∗
7
9
9
1
Ganzha E.I.
n
Dept. Mathematics,
a
J
Krasnoyarsk State Pedagogical University
0
2 Lebedevoi, 89, 660049, Krasnoyarsk, RUSSIA,
2 e-mail: [email protected]
v
2
9 June 1996
0
0
6
0
6 Let us consider the following system introduced in [1] and describing triply orthogonal
9
curvilinear coordinate systems in R3:
/
t
n
i ∂β = λ2 ∂α
- ∂x0 ∂x0
v ∂β = −λ2 ∂α (1)
ol ∂∂(xα1+β) = (β∂x−1 α)∂logλ
s ∂x2 ∂x2
:
v If we will suppose that λ(x ,x ,x )is a given function then (1) becomes a linear system for the
i 0 1 2
X
unknowns α(x ,x ,x ), β(x ,x ,x ). Its compatibility conditions reduce to a single equation
0 1 2 0 1 2
r
a for the function M = logλ (which we will call hereafter a ”potential”):
∂3M ∂M ∂M ∂M ∂M
= cth(M) +th(M) . (2)
∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x
0 1 2 0 1 2 1 0 2
The system (1) has a remarkable property (discovered by G. Darboux, see [1]): if one has
two its solutions {α,β} and {α,β} for a given potential M(x ,x ,x ) then one can obtain
0 1 2
(performing a quadrature) a solution {α ,β } of (1) with a new potential. The formulas of this
1 1
transformation are given by:
M = −M +ln(β/α), (3)
1 0
ξ ξ
α = α− , β = β − , (4)
1 β 1 α
where ξ(α,β,x ) is obtained with a quadrature from the following compatible system
2
∂ξ ∂ξ ∂ξ (α+β)2 ∂ α−β
= β, = α, = − , (5)
∂α ∂β ∂x2 2 ∂x2 α−β!
∗The researchdescribed in this contribution was supported in part by grant RFBR-DFG No 96-01-00050
1
in which one is to suppose that α, β are expressed as functions of α, β, x . Note that α = 1/β,
2 1
β = 1/α also give a solution of (M ) (as we will denote below the system (1) with the potential
1 1
M ), and ξ = − ξ is the corresponding to α , β , α , β solution of (5). This transforma-
1 1 αβ 1 1 1 1
tion is an analogue of the well known Moutard transformation (see. [2, 3, 10]) and was given
in the form (1) by G. Darboux in his monograph [1]. As Darboux had shown ([1]), (3), (4),
correspond to the transformation of the connected to the pair {α,β}, {α,β} of solutions of
(1) triply orthogonal curvilinear coordinate system by inversion. The variation of the second
solution {α,β} correspond to the so called Combescure transformations of the orthogonal co-
ordinate system; composition of inversions and Combescure transformations are equivalent to
compositions of the Ribaucour transformations. Hence we will call the transformation (3), (4)
the Ribaucour-Darboux transformation.
Let us construct starting from some fixed potential M (x ,x ,x ) for which one can find
0 0 1 2
all solutions of (1) (e.g. M = const) a chain of Ribaucour-Darboux transformations (M ) →
0 0
(M ) → (M ) → ... (M ) → .... The potential M will depend (apriori) on 3k functions of 1
1 2 k k
variable — initial data for the solutions (α ,β ) of (M ), s = 0,... ,k − 1. Indeed if we will
s s s
define u = log(β/α) then M = −M +u and the equations for u are given by the elimination
s+1 s
of β from (1) with β = αexp(u):
∂ ∂ u = −e4M+e2u∂ u∂ u+ e2M−eu∂ u∂ M + e2M+eu∂ u∂ M,
0 1 e4M−e2u 1 0 e2M+eu 1 0 e2M−eu 0 1
∂ ∂ u = (e2M−eu)(eu−1)∂ ∂ M + 2(e2M+e2u) ∂ u∂ M − (e2M+e2u) ∂ u∂ u, (6)
0 2 eu(e2M+1) 0 2 (e2M−eu)(eu+1) 0 2 (e2M−eu)(eu+1) 0 2
∂ ∂ u = (e2M+eu)(eu−1)∂ ∂ M + 2(e2M−e2u) ∂ u∂ M − (e2M−e2u) ∂ u∂ u,
1 2 eu(e2M−1) 1 2 (e2M+eu)(eu+1) 1 2 (e2M+eu)(eu+1) 1 2
∂ = ∂/∂x . As one can easily check, (2) give the necessary and sufficient compatibility con-
i i
ditions for (6). Then the known theorem by Darboux [1, p. 335] states that if one will
(0) (0) (0) (0) (0)
fix in a neighborhood of (x ,x ,x ) the initial values ϕ (x ) = u(x ,x ,x ), ϕ (x ) =
0 1 2 0 0 0 1 2 1 1
(0) (0) (0) (0)
u(x ,x ,x ), ϕ (x ) = u(x ,x ,x ) (supposed to be continuously differentiable) then there
0 1 2 2 2 0 1 2
exists the unique solution of (6) with such initial data. So for the Ribaucour-Darboux trans-
formations the potential M depends on M and 3 functions of 1 variable — the initial data
s+1 s
for u .
s
As we have shown in [9], the necessary quadratures (5) may be avoided for (M ), s ≥ 2
s
(”the B¨acklund cube”). For the aforementioned case of the Moutard transformations the so
called ”pfaffian formulas” ([4]) are known as an analogue of the ”wronskian formulas” for
(1+1)-dimensional integrable systems ([5]).
The theory of Ribaucour transformations was extensively studied in the monographs by
Darboux [1], [2] and in numerous papers of his contemporaries [3, 6, 7, 8] for the general
case of the n-dimensional euclidean space. For example L.Bianchi constructed the traditional
((1+1)-dimensional)B¨acklundtransformationforthesystemoftheresonant3-waveinteraction
using Egorov reduction of the equations of triply orthogonal curvilinear coordinate systems
(see [3, 11]). One may ask how wide is the class of potentials for (1) obtainable (via algebraic
operations!) fromagivenpotentialM . Inthetheoryof(1+1)-dimensionalintegrableequations
0
the question of density of the finite gap solutions of the Korteweg-de Vries equation in the class
of all quasiperiodic functions was answered positively. In [10] we have shown that for the case
of the Moutard transformations (such transformations possess the typical properties of (2+1)-
2
dimensional B¨acklund transformations while the Moutard equation itself u = M(x,y)u is
xy
formally (1 + 1)-dimensional) the set of the potentials M obtainable form any fixed M is
k 0
”locally dense” in the space of all smooth functions in a sense to be detailed below.
In this paper we solve the problem of (local) density of M (x ,x ,x ) obtainable from a
k 0 1 2
given initial M (x ,x ,x ) in the space of all solutions of (2). Hence we have proved that the
0 0 1 2
Ribaucour transformation lets us to construct ”almost all” orthogonal curvilinear coordinate
system in R3, consequently the systems of equations describing such coordinate systems ((2)
in particular) shall be considered as ”true” integrable (2+1)-dimensional nonlinear systems.
Note that due to the known theorem by Darboux [1, p. 335] the initial data for (2) are
given by Φ(0,1)(x ,x ) = M(x ,x ,0), Φ(0,2)(x ,x ) = M(x ,0,x ), Φ(1,2)(x ,x ) = M(0,x ,x )
0 1 0 1 0 2 0 2 1 2 1 2
(0) (0) (0)
in a neighborhood of (x ,x ,x ) = (0,0,0).
0 1 2
Theorem 1 Let us fix some M (x ,x ,x ) ∈ C∞ in a neighborhood of (0,0,0). Then for any
0 0 1 2
N = 0,1,2,... one can find K such that for an arbitrary triple system of numbers P(0,1) ,
z1... zk
P(0,2) , P(1,2) , z ∈ {x ,x }, t ∈ {x ,x }, r ∈ {x ,x }, 0 ≤ k ≤ N, the corresponding
t1... tk r1... rk i 0 1 i 0 2 i 1 2
derivatives of the potential M from the chain of the Ribaucour-Darboux transformations coin-
k
cide with P(i,j):
...
p q (0,1) def (0,1)
∂0∂1MK = Pxpxq = Px0... x0x1... x1, (7)
0 1
p q
p q (0,2) def |(0,2){z }| {z }
∂0∂2MK = Pxpxq = Px0... x0x2... x2, (8)
0 2
p q
p q (1,2) def |(1,2){z }| {z }
∂1∂2MK = Pxpxq = Px1... x1x2... x2, (9)
1 2
p q
| {z }| {z }
Obviously we suppose
P d=ef P(0,1) = P(0,2) = P(1,2) = M(0,0,0), P(0) d=ef P(0,1) = P(0,2) = ∂iM (0,0,0),
xi xi xi 0 K
0 0 0
P(1) d=ef P(0,1) = P(1,2) = ∂iM (0,0,0), P(2) d=ef P(0,2) = P(1,2) = ∂iM (0,0,0).
xi xi xi 1 K xi xi xi 2 K
1 1 1 2 2 2
Proof will be given inductively. For N = 0 choose K = 1; from (3) M (0,0,0) =
1
−M (0,0,0) + u (0,0,0). Since (for a given M ) one may set the initial value u (0,0,0) to
0 0 0 0
be arbitrary, and all ∂ku (0,0,0) set to be 0 (the initial Goursat data for (6), one can easily
i 0
show (differentiating M = −M +u ) that for N = 0, K = 1, the following basic induction
1 0 0
proposition holds:
for any N one can find such K that the corresponding derivatives ∂m∂nM , i,j ∈ {0,1,2},
i j K
(i,j)
m+n ≤ N,ofaconveniently chosenM in(0,0,0)coincidewiththegivenP form+n ≤ N:
K xmxn
i j
P(i,j) = ∂m∂nM (0,0,0). Additionally the higher derivatives ∂m∂nM , m+n > N, depend
xmxn i j K i j K
i j
only on P(k,l), p+q ≤ N, in the following way: ∂mM , m > N, coincide with ∂mM (they do
xpxq i K i 0
k l
not depend on the lower derivatives ∂m∂nM , m+n ≤ N); ∂m∂nM , m+n > N, m,n > 0,
i j K 0 1 K
3
depend LINEARLY only upon P(0,1) = ∂p∂qM , p + q ≤ N, p ≤ m; ∂m∂nM , ∂m∂nM ,
xpxq 0 1 K 0 2 K 1 2 K
0 1
(0,1) p q (0,2) (1,2)
m+n > N, m,n > 0, depend upon P = ∂ ∂ M , p+q ≤ N, as well as upon P , P ,
xpxq 0 1 K xkxs xkxs
0 1 0 2 1 2
for k +s ≤ N, s ≤ n.
Inductive step. Suppose that the basic induction proposition is true for the derivatives
of M of orders ≤ N = N , let us prove it for N = N + 1. Let K be the corresponding
K 0 0 0
to N = N number of M . Performing 3N + 3 consecutive Ribaucour-Darboux transfor-
0 K0 0
mations M u−=→u1 M u−=→u2 ... u−=→uN0 M u−=→v1 M u−=→v2 ... u−=→vN0 M u−=→w1
K0 K0+1 K0+N0 K0+N0+1 K0+2N0
M u−=→w2 ... u−=w→N0 M u−=→u(0) M u−=→u(1) M u−=→u(2) M and
K0+2N0+1 K0+3N0 K0+3N0+1 K0+3N0+2 K0+3N0+3
denoting for simplicity Q(x ,x ,x ) = M we have
0 1 2 K0+3N0+3
Q = (−1)N0+1 M −u +u −...±w ∓u(0) ±u(1) ∓u(2) . (10)
K0 1 2 N0
(cid:16) (cid:17)
Let us call the principal derivative of the function u , 1 ≤ i ≤ N , its derivative ∂iu at
i 0 0 i
(0,0,0), and its auxiliary derivative the derivative ∂N0+1−iu at the same point. For v and w ,
1 i i i
1 ≤ i ≤ N , respectively we suppose ∂iv , ∂iw to be the principal and ∂N0+1−iv , ∂N0+1−iw , to
0 2 i 2 i 0 i 1 i
be the auxiliary derivatives at (0,0,0). All the other (nonmixed) derivatives of arbitrary orders
forthese functionswillbeset to0at(0,0,0). Foru(0), u(1), u(2) wesuppose ∂N0+1u(0), ∂N0+1u(1),
0 1
∂N0+1u(2) at (0,0,0) to be the principal derivatives, all the other (nonmixed) derivatives of
2
arbitrary orders for these functions are set to 0 at (0,0,0) (no auxiliary derivatives). Let
us fix the values of u , v , w , u(k) and the values of their auxiliary derivatives at (0,0,0) to
i i i
be ”generic”; more precisely the respective inequalities for them will be specified below, such
(i,j)
inequalities may be obtained using the values of P , k +s ≤ N +1.
xkxs 0
i j
Inside the given step of the main induction (for N = N +1) we will perform an auxiliary
0
induction over m in order to show the validity of the main inductive proposition for ∂m∂nQ for
0 1
all m, n for conveniently chosen principal derivatives of u , u(0), u(1).
i
Firstofallform = 0∂N0+1Q = (−1)N0+1 ∂N0+1M ±∂N0+1u(1) . Thevalueof∂N0+1M at
1 1 K0 1 1 K0
(0,0,0)asweknow fromthemaininduction(cid:16)propositionis equalto(cid:17)∂N0+1M . Consequently one
1 0
may unambiguously determine the principal derivative ∂N0+1u(1) in such a way that ∂N0+1Q =
1 1
Px(N1)0+1. Since ∂1nQ = (−1)N0+1 ∂1nMK0 ± ∂1nuN0+1−n ± ∂1nwN0+1−n , 0 < n ≤ N0, one may
1
uniquely determine the derivativ(cid:16)es ∂nM (they may be chosen ar(cid:17)bitrarily according to the
1 K0
same proposition) using the fixed above auxiliary derivatives of u , v , w in such a way that
i i i
∂nQ be equal to the desired value P(1). Note that due to
1 xn
1
M = (−1)s M −u +... (11)
K0+s K0 1
(cid:16) (cid:17)
the values of M and all their nonmixed derivatives at the origin may be consecutively
K0+s
(using induction on s) found from the derivatives of M and u , v , w , u(s) with respect
K0 i i i
to x 1. So they depend only on P(0,1) . Applying ∂n, n > N + 1, to (10) we see that
= xmxn 1 0
0 1
∂nQ ≡ ∂nM ≡= ∂nM due to the choice of the nonmixed derivatives of v , w , u(s) and the
1 1 K0 1 0 i i
inductive proposition.
TheequalityP(0) = ∂N0+1Qiseasytoobtainasbefore(forP(1) = ∂N0+1Q)choosingthe
xN0+1 0 xN0+1 1
0 1
appropriate principal derivative ∂N0+1u(0). The choice of the lower derivatives ∂nM , n ≤ N is
0 1 0 0
4
analogous to the above case. The proposition about the independence of the higher derivatives
on the lower ones is obvious as well.
Let us choose the principal derivative ∂N0+1u(0)(0,0,0) in such a way that P(2) = ∂N0+1Q.
2 xN0+1 2
2
The required dependence of ∂mQ, m > N +1 is obvious.
2 0
Thus for m = 0 and m = N +1 we have:
0
a) the principal derivatives ∂N0+1u(0), ∂N0+1u(1) and ∂iu , i ≤ m, are already chosen in such
0 1 1 i
a way that the respective equalities (7) p+q ≤ N +1, p ≤ m, are achieved (for Q = M );
0 K
p q
b) the higher derivatives ∂ ∂ Q, p+q > N +1, p ≤ m, and the already defined principal
0 1 0
(0,1)
derivatives depend (according to the construction of Q) only on the values of P ,
xsxt
0 1
s+t ≤ N +1, s ≤ m.
0
We will show the validity of a), b) for 0 < m < N + 1 by an auxiliary induction over m,
0
m ≤ N . In order to perform its step (m = m + 1) apply ∂m0+1∂N0−m0 to (10) taking into
0 0 0 1
consideration the equations (6), (11). We will obtain an expression including in its right hand
p q
side (besides the already defined values of the principal derivatives and ∂ ∂ M , p ≤ m ) the
values of ∂m0+1∂N0−m0M , ∂m0+1∂qM , q < N −m , as well as the on0ly1(soK0far unde0fined)
0 1 K0 0 1 K0 0 0
principal derivative ∂m0+1u (0,0,0) with the coefficient
0 m0+1
exp(4M )+exp(2u )
c = − L m0+1 ∂N0−m0u +F,
m0+1 exp(4M )−exp(2u ) 1 m0+1
L m0+1
L = K +m +1, where F comprises only the defined above values of u , v ,w , their auxiliary
0 0 i i i
derivatives at (0,0,0), the defined during the previous steps of the auxiliary induction principal
derivatives ∂ ui, i ≤ m , i 6= m −1, as well as ∂qM (0,0,0) with q ≤ N .
0 0 0 1 K0 0
Weneedtoshowthat∂m0+1∂qM ,q < N −m ,andtheprincipalderivative∂m0+1u (0,0
0 1 K0 0 0 0 m0+1
may be determined uniquely by the equality to be achieved and by (7) with p + q ≤ N +1,
0
p ≤ m + 1: due to q < N − m the coefficient of the principal derivative in ∂m0+1∂qQ in-
0 0 0 0 1
q
cludes only the initial values of the functions at (0,0,0) and ∂ M which were determined
1 K0
above. We remark now that the value of ∂m0+1∂rM (0,0,0) depend linearly on the values
0 1 K0+s
of ∂m0+1∂qM and ∂m0+1u (0,0,0). Expressing the values of ∂m0+1∂qM linearly via the
0 1 K 0 m0+1 0 1 K0
aforementioned quantities and substituting into the equality we are to establish on this induc-
tive step, we may use the condition of genericity of u (0,0,0) and its auxiliary derivative
∂N0−m0u and conclude that the resulting coefficienmt0o+f1the principal derivative is not equal
1 m0+1
to 0 (this is one of the inequalities constraining them) and consequently we can determine the
principal derivative ∂m0+1u (0,0,0) and afterwards ∂m0+1∂qM , p ≤ N − m , uniquely
0 m0+1 0 1 K0 0 0
in such a way that P(0,1) = ∂m0+1∂N0−m0Q. The proposition about the dependence of
xm0+1xN0−m0 0 1
0 1
∂m0+1∂qQ, m + 1 + q > N + 1, only upon the used P(0,1), s ≤ m + 1, can be easily ob-
0 1 0 0 xsxt 0
0 1
tained from (10) taking into consideration (6) and (11). The auxiliary induction is therefore
completed.
Consider now the equations
P(0,2) = ∂N0+1−m∂mQ(0,0,0),
xN0+1−mxm 0 2
0 2
5
P(1,2) = ∂N0+1−m∂mQ(0,0,0)
xN0+1−mxm 1 2
1 2
simultaneously. We will show the possibility to satisfy them by an appropriate choice of the
principal derivatives ∂iv , ∂iw performing the second auxiliary induction over m.
2 i 2 i
(0) (0,2) (0,1)
Indeed for m = 0 the above equations are already true: P = P = P ,
xN0+1 xN0+1x0 xN0+1x0
0 0 2 0 1
(1) (1,2) (0,1)
P = P = P , as we established above. The proposition about the dependence
xN0+1 xN0+1x0 x0xN0+1
0 1 2 0 1
of the higher order derivatives is trivial in this case. Suppose all these facts to be true for
m ≤ m , we will show their validity for m = m +1. We have (cf. (6), (11))
0 0
P(0,2) =? ∂N0−m0∂m0+1Q = (−1)N0+1 k ∂N0−m0∂m0+1M
xN0−m0xm0+1 0 2 1 0 2 K0
0 2
+a∂m0+1v +b∂m0+1w +F, (cid:16)
Px(N1,02−)m0xm20+1 =?m∂0+1N10−m0∂22m0+1Qm=0+(1−(cid:17)1)N0+1 k2∂1N0−m0∂2m0+1MK0 (12)
1 2
+c∂m0+1v +d∂m0+1w +G, (cid:16)
where F andG comprise2 all tmh0e+t1erms 2includmin0g+1o(cid:17)nly the already found during the previous
inductive steps quantities as well as ∂s∂m0+1M , s < N − m , ∂s∂m0+1M , s < N − m .
0 2 K0 0 0 1 2 K0 0 0
The coefficients k include only the values u , v , w , u(i) at (0,0,0). The coefficients a, d in-
i i i i
clude respectively the auxiliary derivatives ∂N0−m0v , ∂N0−m0w with the coefficients
0 m0+1 1 m0+1
−(exp(2M ) + exp(2v ))/((exp(2M ) − exp(2v ))(exp(2v ) + 1)), −(exp(2M ) −
L m0+1 L m0+1 m0+1 T
exp(2w ))/((exp(2M ) + exp(2w ))(exp(2w ) + 1)), L = K + N + m + 1, T =
m0+1 T m0+1 m0+1 0 0 0
K +2N +m +1, definable via the fixed quantities, these coefficient do not vanish due to the
0 0 0
(i,j)
genericity condition (one can find the corresponding inequalities from P ). Again from the
xkxs
i j
a b
genericity of theauxiliary derivatives one can suppose the determinant to be nonzero at
c d
(cid:12) (cid:12)
(cid:12) (cid:12)
(0,0,0) (again the inequalities for the corresponding auxiliary derivati(cid:12)ves exp(cid:12)ressible in terms
(cid:12) (cid:12)
of P(i,j)). ∂N0−m0∂m0+1M , ∂N0−m0∂m0+1M according to the mai(cid:12)n indu(cid:12)ctive proposition
xkxs 0 2 K0 1 2 K0
i j
(0,1) (0,2) (1,2)
may be found from P , P , P , s ≤ m +1.
xpxq xsxt xsxt 0
0 1 0 2 1 2
The quantities ∂s∂m0+1M , s < N −m , ∂s∂m0+1M , s < N −m , may be found from
0 2 K0 0 0 1 2 K0 0 0
(8), (9) with q = m + 1. Indeed applying ∂s∂m0+1, s < N −m , ∂s∂m0+1, s < N −m , to
0 0 2 0 0 1 2 0 0
(10), one can conclude from (6) that the coefficients of the undefined so far principal derivatives
∂m0+1v , ∂m0+1w due to s < N −m vanish, consequently we (via an induction over s)
2 m0+1 2 m0+1 0 0
unambiguously find ∂s∂m0+1M , s < N −m , ∂s∂m0+1M , s < N −m (their coefficients
0 2 K0 0 0 1 2 K0 0 0
do not vanish due to the genericity condition of the values of u and the other functions at
i
(0,0,0)) in such a way that (8), (9) hold for q = m +1, p+q < N +1.
0 0
Hence by an appropriate (and unique) choice of the principal derivatives ∂m0+1v ,
2 m0+1
∂m0+1w one can satisfy the equalities marked by the question sign in (12). Applying
2 m0+1
to (10) the operators ∂m∂n, ∂m∂n, m+n > N +1, n ≤ m +1, we get the necessary result on
0 2 1 2 0 0
the dependence of the higher derivatives ∂m∂nQ, ∂m∂nQ upon P(i,j).
0 2 1 2 xkxs
i j
Consequently the main inductive proposition and the Theorem are proved.
Remark. One may consider of interest to generalize this result for the case of the Ribaucour
transformationsoftheorthogonalcurvilinear coordinatesystems inthen-dimensional euclidean
6
space ([1, 3]). We shall note that the system describing such coordinates is an overdetermined
system with the initial (Goursat like) data given by n(n−1)/2 functions of twovariables for ar-
bitrary n. Consequently we have an (2+1)-dimensional system possessing a (2+1)-dimensional
B¨acklund transformation — the Ribaucour transformation. Unfortunately no representation
analogous to (1) is known for this case. Nevertheless one may conjecture the hypothesis about
the completeness of the Ribaucour transformations also in the n-dimensional case.
References
[1] Darboux G. Lec¸ons sur les syst`emes orthogonaux et les coordonn´ees curvilignes, 2-`eme
ed., Paris, 1910.
[2] Darboux G. Lec¸ons sur la th´eorie g´en´erale des surfaces et les applications g´eom´etriques du
calcul infinit´esimal, t. 1–4, Paris, (1887–1896).
[3] Bianchi L. Lezioni di geometria differenziale, 3-a ed., v. 1–4, Bologna, 1923–1927.
[4] Athorne C., Nimmo J.J.C. On the Moutard transformation for integrable partial differen-
tial equations, Inverse Problems, 1991, v. 7, p. 809–826.
[5] Matveev V.B., Salle M.A. Darboux transformations and solitons. Springer-Verlag, 1991.
[6] Tzitz´eica G. Sur les congruences cycliques et sur les syst`emes triplement conjugu´es, Ann.
Sci. E´cole Norm. Sup. 1899, v. 16, p. 137–192.
[7] Demoulin A. Recherches sur les syst`emes triples orthogonaux, M´emoirs Soc. Roy. Sciences
Li`ege, 1921,v. 11, 98 p.
[8] Guichard C. Sur les syst`emes orthogonaux et les syst`emes cycliques, Ann. Sci. E´cole Norm.
Sup. 1897, v. 14, p. 467–516, 1898, v. 15, p. 179–227, 1903, v. 20, p. 75–132, 181–289.
[9] GanzhaE.I., TsarevS.P.OnsuperpositionoftheautoB¨acklundtransformationsfor(2+1)-
dimensional integrable systems, submitted to [email protected]
[10] Ganzha E.I. On completeness on the Moutard transformations, submitted to solv-
[email protected]
[11] Tsarev S.P. Classical differential geometry and integrability of systems of hydrodynamic
type, in: Proc. NATO ARW ”Applications of Analytic and Geometric Methods to Nonlin-
ear Differential Equations” (ed. P. Clarkson), NATO ASI Series, Kluwer, 1992, p. 241–249.
7
