A construction of negative flows for integrable systems based on the Lax representation and squared eigenfunctions is proposed. Examples considered include the Boussinesq equation and its reduction to the Sawada–Kotera and Kaup–Kupershmidt equations; one of the Drinfeld–Sokolov systems and its reduction to the Krichever–Novikov equation.

1.
J.
Schiff
, “
The Camassa–Holm equation: A loop group approach
,”
Physica D
121
(
1–2
),
24
43
(
1998
).
2.
A. N. W.
Hone
, “
The associated Camassa–Holm equation and the KdV equation
,”
J. Phys. A: Math. Gen.
32
(
27
),
L307
L314
(
1999
).
3.
A. G.
Meshkov
and
V. V.
Sokolov
, “
Hyperbolic equations with third-order symmetries
,”
Theor. Math. Phys.
166
(
1
),
43
57
(
2011
).
4.
C.
Rogers
and
W. K.
Schief
,
Bäcklund and Darboux Transformations. Geometry and Modern Applications in Soliton Theory
(
Cambridge University Press
,
Cambridge
,
2002
).
5.
A. M.
Kamchatnov
and
M. V.
Pavlov
, “
On generating functions in the AKNS hierarchy
,”
Phys. Lett. A
301
(
3–4
),
269
274
(
2002
).
6.
H.
Aratyn
,
J. F.
Gomes
, and
A. H.
Zimerman
, “
On negative flows of the AKNS hierarchy and a class of deformations of a bihamiltonian structure of hydrodynamic type
,”
J. Phys. A: Math. Gen.
39
(
5
),
1099
1114
(
2006
).
7.
V. E.
Adler
, “
Negative flows and non-autonomous reductions of the Volterra lattice
,”
Nonl. Math. Phys.
2024
.
8.
V. E.
Zakharov
and
A. B.
Shabat
, “
A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I
,”
Funct. Anal. Appl.
8
(
3
),
226
235
(
1975
).
9.
M. J.
Ablowitz
and
R.
Haberman
, “
Resonantly coupled nonlinear evolution equations
,”
J. Math. Phys.
16
(
11
),
2301
2305
(
1975
).
10.
V. G.
Drinfeld
and
V. V.
Sokolov
, “
Lie algebras and equations of Korteweg–de Vries type
,”
J. Sov. Math.
30
(
2
),
1975
2036
(
1985
).
11.
M.
Gürses
,
A.
Karasu
, and
V. V.
Sokolov
, “
On construction of recursion operators from Lax representation
,”
J. Math. Phys.
40
(
12
),
6473
6490
(
1999
).
12.
A. S.
Fokas
and
R. L.
Anderson
, “
On the use of isospectral eigenvalue problems for obtaining hereditary symmetries for Hamiltonian systems
,”
J. Math. Phys.
23
,
1066
1073
(
1982
).
13.
S. Y.
Lou
and
M.
Jia
, “
From one to infinity: Symmetries of integrable systems
,” arXiv:2309.06729 (
2023
).
14.
K.
Sawada
and
T.
Kotera
, “
A method for finding n-soliton solutions of the KdV equation and KdV-like equations
,”
Prog. Theor. Phys.
51
(
5
),
1355
1367
(
1974
).
15.
D. J.
Kaup
, “
On the inverse scattering problem for cubic eigenvalue problems of the class ψxxx + 6x + 6 = λψ
,”
Stud. Appl. Math.
62
(
3
),
189
216
(
1980
).
16.
A.
Degasperis
and
M.
Procesi
, “
Asymptotic integrability
,” in
Symmetry and Perturbation Theory, Proceedings of the 2nd International Workshop, Rome, Italy, December 16–22, 1998
, edited by
A.
Degasperis
, and
G.
Gaeta
(
World Scientific
,
Singapore
,
1999
), pp.
23
37
.
17.
A.
Degasperis
,
D. D.
Holm
, and
A. N. W.
Hone
, “
A new integrable equation with peakon solutions
,”
Theor. Math. Phys.
133
(
2
),
1463
1474
(
2002
).
18.
I. M.
Krichever
and
S. P.
Novikov
, “
Holomorphic bundles over algebraic curves and nonlinear equations
,”
Russ. Math. Surv.
35
(
6
),
53
79
(
1980
).
19.
J. P.
Wang
, “
A list of 1 + 1 dimensional integrable equations and their properties
,”
J. Nonlinear Math. Phys.
9
(
Supplement 1
),
213
233
(
2002
).
20.
V. E.
Adler
and
A. B.
Shabat
, “
Toward a theory of integrable hyperbolic equations of third order
,”
J. Phys. A: Math. Theor.
45
,
395207
(
2012
).
21.
M.
Gürses
and
A.
Karasu
, “
Integrable KdV systems: Recursion operators of degree four
,”
Phys. Lett. A
251
,
247
249
(
1999
).
22.
S. I.
Svinolupov
,
V. V.
Sokolov
, and
R. I.
Yamilov
, “
On Bäcklund transformations for integrable evolution equations
,”
Sov. Math. Dokl.
28
,
165
168
(
1983
).
23.
V. E.
Adler
, “
Nonautonomous symmetries of the KdV equation and step-like solutions
,”
J. Nonlinear Math. Phys.
27
(
3
),
478
493
(
2020
).
24.
V. E.
Adler
and
M. P.
Kolesnikov
, “
Non-autonomous reductions of the KdV equation and multi-component analogs of the Painlevé equations P34 and P3
,”
J. Math. Phys.
64
,
101505
(
2023
).
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