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John Weiss
Applied Mathematics Group
49 Grand View Road
Arlington, MA 02174
Previously, we have found a factorization of the (1+1)-dimensional Toda lattice by the periodic fixed points of its' Bäcklund transformations. The Toda flow is realized by two commuting, one dimensional Hamiltonian flows. By a result of Konopelchenko, the Laplace-Darboux transformation is a Bäcklund transformation for the (2+1)-dimensional Boiti, Leon and Pempinelli (BLP) equation. A periodic fixed points of the Laplace transformation is an invariant manifold of the BLP flow. This manifold is determined by solutions of the (1+1)-dimensional Toda lattice equations. From these results we find that the (2+1) BLP flow is factored by three commuting, one dimensional Hamiltonian flows that are the periodic fixed points of its' Bäcklund transformations.
The Two dimensional Toda lattice equations were first studied in the works of Laplace, Moutard and Darboux [1] in connection with their classification of surfaces and factorizations of linear differential operators. For instance, Moutard essentially solved the free end Toda lattice in the form of Wronskian determinants [1] . The two dimensional Toda lattice was derived by Laplace in the eighteenth century through results relating factorizations of linear differential operators with certain gauge invariants. This Laplace transformation was proposed by Darboux as a fundamental method for the classification of certain surfaces in space related as focal, or caustic surfaces.
By a result of Konopelchenko [2] , the BLP equation is factored by the periodic fixed points of the Laplace transformation onto an invariant manifold determined by solutions of the two dimensional periodic Toda lattice.
Previously, we have found [3] an infinite set of distinct Bäcklund transformations for the two dimensional Toda latttice. Although simple to describe the resulting systems of integrable ordinary differential equations have a rich structure that depends strongly on the length and number theoretic properties of the period of the lattice. We emphasize that the time-space dependence of the lattice is factored by commuting, finite dimensional hamiltonian flows. Surprisingly, these systems arise naturally from the periodic fixed points of the simple Bäcklund transformations.
Using the above set of results we find a factorization of the BLP equation by three commuting, one dimensional Hamiltonian flows that are the periodic fixed points of its' Bäcklund transformations.
By way of introduction we present the relevant results for the KdV system and the derivation of the Toda lattice from the Laplace transformation of focal surfaces. Then, we show how a simple generalization of the KdV result obtains the set of Bäcklund transformations for the Toda lattice.
We then present the factorization of the BLP equation by the periodic fixed points of its' Bäcklund transformations.
The Korteweg-deVries equation,
has the Bäcklund transformation [4]
where 
and
Equation (1) is somtimes known as the singular manifold equation for the KdV equation and itself has the two Bäcklund transformations [4] , [6]
and
where, for both transformations,
The expression
is the Schwarzian derivative, which is invariant under the Moebius group (2) [4] , [5] .
By itself, transformation (2) is a point symmetry that does not lead to new forms of solution, and transformation (3) by itself is in involution. The effective Bäcklund transformation (BT) for (1) is the composition of (2) and (3). We find that [6] :
is a BT for (1). The periodic fixed points of the BT are defined by equations (4) and (5) with:
The periodic fixed points continue to define a strong BT for (1). That is, the integrability conditions
continue to inply that 
satisfy
(1), and, by the periodicity 
,
the set
are
solutions of (1).
We have found [6] that if
then
The KdV and Boussinesq systems are instances of the general system in component form [7]
where
. The KdV
systems correspond to p=1 and the Boussinesq to p=2. Let
the circulant forward shift matrix [8] be
In the N-vector form equations (6) are
with
The casimir integrals of (6) correspond to the null vectors of B. The null vectors of A produce the constraints.
Associated with the principal casimir, for any N
we find the principal integrals of (6)
,where 
and
The systems (7) have a Hamiltonian structure
,where 
The higher-order equations associated with the integrals (8) are
When A is invertible, then
is an antisymmetric circulant matrix.
We have the systems
and
where
is the co-symplectic form.
With p=1 the flow of the KdV equation is factored by the commuting, hamiltonian systems [6]
Now, Darboux [1] has shown that the parameters
(x,y) for surfaces in three dimensions can be defined so
the coordinates
of
the surface satisfy a partial differential equation of the form:
,where (a,b,c) are functionals of the first fundamental form in the (x,y) parameters.
Under the gauge transformation 
,
the form of (13) is preserved and:
are invariant.
The Laplace transformation of a surface is a partial factorization of (13) in terms of the invariants [1] .
Equations (14) imply that z satisfy (13) and 
satisfy
the system
where
From (15) the Laplace transformation of the invariants is
Darboux [1] studied the periodic fixed points of the Laplace transformation and found that these surfaces are related as a sequence of focal surfaces. From (16), the periodic fixed points are
, where 
and
n is the order of the fixed point. The substitution
obtains the two dimensional periodic Toda lattice
A result of Konopelchenko [2] exhibits the Laplace transformation as a Bäcklund transformation for the BLP equation [9] . For this purpose it is important to note that the condition b=0 is invariant under the Laplace transformation. The Lax pair for the BLP equation can be written as
and, in terms of the invariants of the Laplace transformation, 
and 
,
These equations are equivalent to the BLP equation.
The periodic fixed points of Laplace transform ,with 
and 
,
factor the flow into the two dimensional periodic Toda lattice
and the system
Now, the Bäcklund transformations for the Darboux equations
(17) and the Toda lattice equations (18) was found in reference [3]
. With reference to systems (11) and (12), without loss of generality normalize
the casimir, 
,
and set
,where 
.
Then, let 
and
find that (19), (20) imply
,where 
.
To see this let 
and
find
where
It can be shown that
Let 
and
find (24).
When p=1 (25) are the Toda lattice of period N. If N
and p are relatively prime (25) is again a Toda lattice of length
N. If N=mp (25) is p distinct lattices of length
m. When N and p have common factors there is one lattice
for each distinct orbit of translation by 
.
In all cases the set of fields 
are
directly related to the set of invariants 
.
When A is not invertible we find for equations (9) and (10) a similar
connection with the Toda lattice. In this case one must take into account
the constraints that apply to these systems to obtain a valid correspondence.
See, for instance, reference [7] .
Consideration of the form of (23), (24) and the possible relations between
p and N determine that for a lattice of fixed length m
there will exist an infinite sequence of distinct Bäcklund transformations.
For instance, we have a Bäcklund transformation for a lattice of length
m when N=pm for 
.
Systems (23) and (24) have a rich structure. As a Bäcklund transformation for a Toda lattice of fixed length these systems relate flows from different hierarchies of equations with the flows in the sequence through the Toda lattice. In other words, as described by (23) and (24) the hierarchies of flows through KdV, Boussinesq, etc. are interelated.
The BLP system is related to the above construction when p=1. In this case we have
and
The periodic fixed points of the KdV type, p=1, Bäcklund transformations factor the flow of the BLP system into three commuting hamiltonian systems
where
From equations (26) and (27)
and substitution in equation (22) obtains
Equations (26) and (28) imply
Substitution into equation (30), using (26), obtains
This equation may be written in the form
By the dual hamiltonian structure found in reference [6] , eq. (2.53), we obtain equation (29) and by the results of reference [6] the flows (27), (28) and (29) commute.
Surprisingly, the factorization of the Toda lattice for p=2 does not lead to a corresponding factorization of the BLP equation. It is possible that tha case p=1 is unique in this regard.