# A Dataless FaceSwap Detection Approach Using Synthetic Images.

Anubhav Jain, Nasir Memon, Julian Togelius
New York University
{aj3281, nm1214<EMAIL_ADDRESS>
###### Abstract

Face swapping technology used to create ”Deepfakes” has advanced significantly
over the past few years and now enables us to create realistic facial
manipulations. Current deep learning algorithms to detect deepfakes have shown
promising results, however, they require large amounts of training data, and
as we show they are biased towards a particular ethnicity. We propose a
deepfake detection methodology that eliminates the need for any real data by
making use of synthetically generated data using StyleGAN3. This not only
performs at par with the traditional training methodology of using real data
but it shows better generalization capabilities when finetuned with a small
amount of real data. Furthermore, this also reduces biases created by facial
image datasets that might have sparse data from particular ethnicities. To
promote reproducibility the code base has been made publicly available
111https://github.com/anubhav1997/youneednodataset .

##  1 Introduction

Numerous approaches have been proposed to create photo-realistic deepfakes in
recent years. With every new approach that is proposed, the quality of the
deepfakes is getting better; we will likely eventually reach a point where
humans are unlikely to distinguish them from real images or video[24]. Most
current approaches make use of face-swapping models where the face of a person
is replaced with that of another in an image or video. While current
approaches to detecting deepfakes have been promising they require a large
dataset of facial images. This can be challenging as publicly available
datasets have extremely strict licenses or in some cases do not have the
rights for use of the data cleared - such as the FakeAVCeleb, FF++, Celeb-DF,
and UADFV to name a few. Some of these datasets have directly used videos from
YouTube or other such public spaces without obtaining proper consent from the
people that appear in the videos. Published datasets that have claimed to use
images of celebrities scraped from public forums have often included data from
”ordinary” individuals [23] and have hence been discontinued due to such
reasons [22]. This leads to a need to move away from such approaches that
require large amounts of privacy-sensitive data.

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Figure 1: Examples of images generated using StyleGAN3 [18].

In fact, biometric data privacy is now considered a fundamental right and the
collection and usage of such data is regulated under the law. For example, in
2017 the Indian supreme court ruled that privacy is a fundamental right of its
residents and further emphasized that the protection of biometric data is of
utmost importance. Recently, we have seen regulations such as the General Data
Protection Regulation by the European Union that have brought the usage of
biometric data under strict data protection laws. In the United States,
similar steps have been taken. In California, the California Consumer Privacy
Act (CCPA) and California Privacy Rights Act (CPRA) were put in place for this
very purpose. In 2008, Illinois set forth the Biometric Information Privacy
Act (BIPA), which has so far also been adopted by the states of Washington and
Texas. Companies such as Facebook [26], Google [3], and Shutterfly [1] have
come under the scanner for their usage of facial images of their users under
the BIPA law. On a national level, the National Biometric Information Privacy
Act of 2020 (Senate Bill 4400) was introduced, if passed it would further
strengthen the regulations on such data [2].

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Figure 2: Examples of face swaps created using SimSwap and Sberswap.

There has also been an ongoing debate on whether researchers should be legally
allowed to use publicly available biometric data and especially faces [12].
While it is still not illegal to do so, it is often considered to be an
ethical issue. This stems from the fact that the users appearing in these
datasets are neither informed about their data being used nor has their
consent been taken beforehand. If a user deletes their data from a public
forum or wishes to do so, the machine learning models built using their data
cannot be tracked and machine unlearning of this data which is no longer
publicly available is close to impossible. Moreover, there is no way of
tracking the datasets that are using data that the user no longer deems to be
public. These ethical and legal conundrums have resulted in approaches for
building vision models that move away from using data that contain biometric
data of individuals.

In this paper, we propose a dataset-free approach for training a face-swap
detection model that will help in bypassing such key privacy concerns by
utilizing images of people that don’t exist. We do this by generating fake
identities and using these to create deepfakes or face swaps.

Since we create face swaps on the fly a major concern that arises with such an
approach is that we assume that we have access to the face-swapping model in
question. While this may seem to hamper its applicability in the real-world
scenario, we show that this detection model built upon a face-swapper is
generalizable to unseen face-swapping algorithms. We further emphasize this by
showing that the detection model learns interpretable features through
visualization using Captum [21]. We observe that the model performs the best
when it is fine-tuned with a small sample of real data to take care of the
domain shift between synthetic and real data.

Finally, we show that the deepfake detector trained on real-world data is
biased against certain ethnicity. This has serious implications where
deepfakes pertaining to this ethnicity is less likely to be detected. And as
is often the case this arises due to biases existing in the dataset that is
used to train the classification model. Our approach eliminates this dataset
induced bias by using synthetic images. We show empirically in the paper that
training a face swap detector using synthetic data helps in removing biases in
the model architecture. A simple intuition is that synthetic datasets or GAN
models give an additional level of control over the image we want. Even simply
random sampling from the GAN latent space is powerful enough to reduce the
bias significantly as we will show in this paper.

##  2 Related Work

###  2.1 Deepfake and Face Swapping Detection

There has been a lot of recent interest in the detection of deepfakes and
face-swapped images. Face-swapping methodologies can be divided into
traditional face-swapping methods that use software such as Adobe Photoshop
and the more recent GAN-based face-swapping models. In this paper, the focus
is on the detection of the latter. Ding et al. [7] proposed a transfer
learning-based approach to detect face swaps that were created using
Autoencoder GAN and Nirkin’s method. The dataset that the authors created
consisted of approximately 420,000 images. The frequency-domain has been used
for the detection of face swaps in conjunction with a CNN-based network [20].
Guan et al. [10] proposed an approach to detect face swaps based on the
inconsistencies in the 3D facial shape and appearance-based features. The
authors tested the model on the FaceSwap and Deepfakes subset of the
FaceForensics++ dataset. Other researchers have also proposed approaches that
improve the generalizability of these classifiers such as Jiang et al. [16].
However, their work requires the corresponding real face as a reference for
prediction - which may not always be feasible to acquire. This also raises
certain privacy concerns.

###  2.2 Synthetic Datasets

Generative adversarial networks have widely been used to create realistic
synthetic imagery. In the past couple of years, GAN-generated images have
become more and more realistic and have been successful in deceiving human
beings. Now the question arises that since these images are indistinguishable
from real images, can they also be used to train deep classifiers? Grosz et
al. [9] showed that fingerprint spoofing classification benefits from the
usage of synthetic GAN-generated fingerprints. They trained the spoofing
classifier first only with generated images and further fine-tuned it using
real images which shows better performance as compared to only training using
real images. Jahanian et al. [13] used generated images as a source for
multiview representation learning. They show that the learned representations
are as good or sometimes even better than the ones learned using real data.
Moreover, using contrastive learning, they were able to easily find positive
pairs by generating images from nearby latent vectors.

Other benefits of using synthetic datasets have also been studied. Jaipuria et
al. [15] showed that using synthetic data augmentation they were able to
improve the model generalizability in the cross-dataset scenario by
introducing more diversity in the training dataset. While these studies
comprehensively show that there is a reasonable advantage to using synthetic
data, they do not discuss the inherent biases introduced in the models due to
class imbalance, and under-representation of particular ethnicities, genders,
or racial groups.

##  3 Datasets

In the paper we evaluate the trained face swap detection models on three
datasets - Flickr Faces High Quality (FFHQ), CelebA-HQ, and the Amsterdam
Dynamic Facial Expression Set (ADFES).

The Flickr Faces HQ dataset [19] contains 70,000 high quality 1024x1024 real
closeup facial images. The images were scrapped from Flickr and thus have some
distribution amongst various age groups, ethnicities, and backgrounds. The
images were further processed by aligning and cropping them using dlib. We
split the dataset into train and test sets with 30,000 images in the test set
and the remaining 40,000 in the train set. For validation, we have utilized a
small subset of images from the train test.

The CelebA-HQ dataset [17] is higher quality version of the CelebA dataset of
resolution 1024x1024. As the name suggests the dataset contains images of
celebrities. The images have variable backgrounds and are diverse in terms of
ethnic groups, age, and gender. We utilize the entire dataset for testing our
models. Given the diversity in the dataset, we also use it for experimentation
on understanding the biases present in the detection models.

The Amsterdam Dynamic Facial Expression Set (ADFES) [8] is a smaller dataset
that contains 648 images from 22 models showcasing nine facial expressions:
fear, anger, contempt, disgust, joy, sadness, pride, embarrassment, and
surprise. The images in the dataset are of size 576x720 and thus had to be
resized to 1024x1024 using bilinear interpolation. Moreover, the dataset
contains only Northern European and Mediterranean models. The entire dataset
is utilized for testing.

##  4 Proposed Approach

In this section, we describe the proposed pipeline of using synthetic images
to train a face-swapping detection algorithm. To fairly evaluate the
advantages of using synthetic images over using real images from the FFHQ
dataset for training the classifier we have kept all other parameters
consistent including preprocessing steps along with optimization parameters
batch size and learning rate.

Table 1: Evaluation of the Xception trained only on synthetic data (row 1), real data from FFHQ dataset (row 2), and synthetic data with finetuning on FFHQ dataset (row 3) on the FFHQ, ADFES, and CelebA-HQ dataset. The classifier is trained to detect images from one face swapper and is tested in a cross-modal scenario on another.  Train Type | Train Swap Model | Test Swap Model | FFHQ | ADFES | CelebA-HQ
---|---|---|---|---|---
Trained on FFHQ | Simswap | Simswap | 0.9973 | 1.0000 | 0.7437
|  | Sberswap | 0.9972 | 1.0000 | 0.7437
| Sberswap | Simswap | 0.9973 | 0.9979 | 0.9341
|  | Sberswap | 0.9974 | 0.9979 | 0.9341
Synthetic Data | Simswap | Simswap | 0.9972 | 0.9979 | 0.8821
|  | Sberswap | 0.8802 | 0.9353 | 0.8509
| SberSwap | Simswap | 0.9913 | 0.9916 | 0.8221
|  | Sberswap | 0.9921 | 0.9916 | 0.8236
Finetuned on FFHQ | Simswap | Simswap | 0.9974 | 1.0000 | 0.9167
|  | Sberswap | 0.9971 | 1.0000 | 0.9164
| Sberswap | Simswap | 0.9955 | 0.9979 | 0.9979
|  | Sberswap | 0.9973 | 1.0000 | 0.9877
Table 2: Evaluation of the biases on the model trained with real data vs the
models trained with synthetic data when testing on the CelebA-HQ dataset using
the metrics Accuracy Difference (AD), Difference in Rejection Rate (DRR), and
Difference in Acceptance Rate (DAR).

Train Type | Train Swap | Test Swap | Ethnicities | Gender | Age
---|---|---|---|---|---
| Model | Model | AD | DRR | DAR | AD | DRR | DAR | AD | DRR | DAR
Trained on | Simswap | Simswap | 0.2233 | 0.2385 | 0.0181 | 0.1581 | 0.1555 | 0.0064 | 0.3384 | 0.5409 | 0.0069
FFHQ |  | Sberswap | 0.2573 | 0.2893 | 0.0181 | 0.1402 | 0.1236 | 0.0070 | 0.3388 | 0.6686 | 0.0103
| Sberswap | Simswap | 0.0933 | 0.1656 | 0.0160 | 0.0589 | 0.1030 | 0.0054 | 0.2828 | 0.9044 | 0.1429
|  | Sberswap | 0.1053 | 0.2172 | 0.0142 | 0.0542 | 0.0865 | 0.0056 | 0.2813 | 0.8983 | 0.1430
Finetuned | Simswap | Simswap | 0.0729 | 0.1281 | 0.0164 | 0.0384 | 0.067 | 0.0054 | 0.3703 | 0.6646 | 0.0133
on FFHQ |  | Sberswap | 0.0982 | 0.1923 | 0.0159 | 0.0331 | 0.0483 | 0.0047 | 0.3676 | 0.8624 | 0.0100
| Sberswap | Simswap | 0.0180 | 0.0221 | 0.0184 | 0.0069 | 0.0151 | 0.0043 | 0.2223 | 0.9816 | 0.1250
|  | Sberswap | 0.0198 | 0.0334 | 0.0154 | 0.0086 | 0.0117 | 0.0058 | 0.2223 | 0.9818 | 0.1250

Figure 3: Overview of the proposed approach to detect face-swapped images
using synthetically generated images.

The pipeline contains 3 major steps as shown in figure 3. These are the
generation of synthetic data, face swapping using the generated data, and
finally classification of the face swaps.

###  4.1 Data Generation

We generated synthetic images using a pre-trained StyleGAN3[18] model. The
StyleGAN3 model was trained on the Flickr-Faces High-Quality [19] dataset to
generate 1024x1024 closeup facial images. Figure 1 shows some examples of the
generated images. Instead of creating a synthetic dataset [13] which would
require a high amount of storage space we generate batches of images and use
this for training. To ensure that the faces we generate are dissimilar we use
a random vector generated using a different seed value for every batch.
Moreover, we randomly select the truncation-psi value between $[0,1]$ using a
normal distribution.

###  4.2 Face Swapping

The generated images from StyleGAN3 are further used for creating face swaps.
The face swapper takes in a batch of GAN-generated images and generates $N-1$
number of swapped images by swapping the $i-th$ image with the $(i+1)-th$
image.

We used two methodologies to swap faces - SimSwap [4] and SberSwap [5]. Both
of these are GAN-based models that have been trained on the VGGFace2-HQ and
VGGFace2 datasets respectively.

The SimSwap architecture consists of three parts - the encoder, ID injection
module (IIM), and the decoder. The ID injection module transfers the identity
of the source image onto the target. The network is trained with an identity
loss to ensure that the identity of the source image is generated. The network
ensures that other facial attributes such as expression and gaze direction are
transferred.

The SberSwap algorithm is an improvement over the FaceShifter network [5].
They make use of the AEI module combined with an adaptation of reconstruction
loss proposed by SimSwap. They have an additional super-resolution block that
makes the resulting images more realistic. They also added an eye loss, that
was specifically made to ensure the consistency of the eye. The discrepancies
in the eye and iris shape have been discussed in detail by previous
researchers as being an easy way to spot deepfakes [11].

###  4.3 Classification Network

Finally, for the classification network, we use the Xception network [6].
Authors of the FaceForensics++ dataset [24] found this to be the best
performing model on their dataset. We have used the same hyperparameters as in
the original paper for consistency.

The Xception network is trained for binary classification with a categorical
cross-entropy loss function to distinguish between the face swapped and the
original images. We changed the input and output classification layers from
the original ImageNet model architecture. For training the classifier binary
classification model, the synthetically generated images are considered as the
”real class” and the face swapped images are considered as the ”fake class”.
While the synthetic media isn’t truly a ”real class” and is often considered
to be fake, in this particular case the deep learning model will learn
distinguishing features between the two classes. Since both classes have
synthetically generated images, the distinguishing features would come from
the face swapper.

As a baseline model, we train the XceptionNet using only real images from the
training set of the FFHQ dataset instead of using synthetically generated
images. For fair comparison, only the training data has been changed, while
keeping the model architecture and other training dynamics the same. The two
models are trained using a batch size of 12 - which was the largest batch that
could fit on an RTX-8000 GPU node. Due to the large size of the FFHQ dataset,
we limited the number of steps per epoch to 2000 for both the models that were
trained using real data and synthetic data.

##  5 Results

###  5.1 Detection Performance

The classification performance of the models trained on real and synthetic
data is tested both in the seen and unseen scenarios with respect to the face
swap model and the dataset. We performed two sets of cross-modality
experiments to test the efficiency of the proposed approach. First, we tested
the model against an unseen face swap attack. For example, a model trained
only on face swaps generated from SimSwap is tested on face swaps generated
from SberSwap. Secondly, there might be some information leakage in the case
of the FFHQ dataset through the StyleGAN3 model which was trained on this
dataset, thus we test the model on two other datasets that have been collected
in completely different settings and backgrounds. In fact images from the
ADFES dataset had to be resized without maintaining the aspect ratio. A model
that is able to generalize in such a scenario is immune to resizing-based
image transformations. This is typically not the case with models that learn
noise level differences and artifacts.

We observed that simply training on GAN-generated images led to good
performance on the FFHQ and the ADFES dataset ( 99% accuracy for both) and
even generalized to a different face-swapping method than it was originally
trained on. However, the accuracy results for the CelebA-HQ dataset were lower
as compared to the model trained on real FFHQ data. The model trained on real
data with both the swapping and testing model being SberSwap had a detection
accuracy of 93.41% and for the same setting, the model trained on synthetic
data had only 82.36% accuracy. While there were also scenarios where the model
trained on synthetic data performed better than the model trained on real
data, this showed that there was still a need for overcoming the domain shift
to perform better than the model solely trained on real data. For this
purpose, we finetune the model trained solely on synthetic data with only
12,000 real images from FFHQ. This finetuned model performed better than the
model solely trained on synthetic data in all combinations of datasets and
face-swapping model - generalizing well across different datasets and face-
swap technologies.

###  5.2 Bias

We hypothesize that the models trained on synthetic data would be fairer or
less biased. Since models trained on real data exhibit similar biases to the
datasets they are trained on and in this case, the absence of a dataset allows
an extra layer of control to mitigate bias. Our empirical results show that
even simply random sampling images from a GAN model leads to significantly
lesser bias in the resultant classification model.

To test our hypothesis we utilize three metrics for evaluation of the post-
training model biases - accuracy difference (AD), the difference in acceptance
rate, and the difference in rejection rate. Accuracy difference is the maximum
different between the classification accuracy of different facets in a set of
demographics $\\{d_{1},d_{2},...,d_{n}\\}\in D$ (equation 1). DAR and DRR have
similarly been defined as the maximum difference in acceptance and rejection
rates respectively (eq. 2 and 3). Here acceptance rate is defined as the ratio
of true positive predictions to the observed positives and the rejection rate
is defined as the ratio of the true negative predictions to the observed
negatives.

$AD=max_{i,j}|ACC_{i}-ACC_{j}|\forall i,j\in D$ (1)

$DAR=max_{i,j}|AR_{i}-AR_{j}|\forall i,j\in D$ (2)

$DRR=max_{i,j}|RR_{i}-RR_{j}|\forall i,j\in D$ (3)

For this work, we have categorized the people based on their ethnicity/race,
age, and gender. DeepFace model [25] was used for categorization of the image
into these subgroups based on the predicted race, age, and gender. We have
considered six ethnic groups of people - white, Asian, Indian, black, Latino
Hispanic, and middle eastern based on the predictions of DeepFace. The people
are classified into two genders - male and female and we have quantized the
estimated age into buckets of 10 years.

We observed significant improvement in the biases of the model fine-tuned on
real data vs the model solely trained on real data. The results have been
summarized in table LABEL:tab:bias. From this, we observe that the accuracy
difference for the former is roughly 10 times lesser than the latter in most
cases. Similarly, DRR is about half and DAR is almost the same. The only
anomaly is in the case of biases in age groups where the model trained on real
data is slightly better in terms of AD and DRR. However, it is important to
note that particular age groups such as ’10-20’ and ’70-80’ which had only 16
and 9 samples in the CelebA-HQ dataset have largely influenced these values.
Thus, the major focus should remain on the values pertaining to the
subcategories of ethnicity and gender where there are a sufficient number of
samples per subcategory ($>500$).

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Figure 4: Observed Saliency plots for the XceptionNet models trained on
synthetic data (top row) and real data (bottom row) showing the gradient
magnitudes.

###  5.3 Interpretability

Finally, we analyze the model trained on synthetic data with the model trained
on real data using Captum [21] to observe the areas the model focuses on for
making predictions. The objective behind this experiment is to observe whether
these models are learning meaningful and human interpretable differences
brought by the face swapper or if it is focusing on noise-level artifacts. We
utilize two methodologies provided by Captum - Saliency and Occlusion. The
former computes the gradients with respect to the class index and transposes
the output for visualization. While for the latter a sliding window is used to
iteratively occlude part of the image. The change in the model output is
quantified and visualized.

Figure 5 depicts the Saliency plots for the two models. It can be seen that
the model trained on synthetic data has more focused regions that clearly
align with the human face attributes such as the eyes, nose, and mouth.
However, for the model trained on real data, the higher valued gradients are
more spread around the face and some background regions.

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Figure 5: Examples regions that the model focuses on for making the decisions
for both using synthetic data (top row) and real data for training (bottom
row).

Similarly, for the Occlusion plots in figure 4, it can be observed that the
model trained on synthetic data learns more interpretable and generalizable
features. The model trained only on real data focuses on the entire face
region along with some background regions while on the other hand, the model
trained on synthetic data learns to specifically focus on facial attributes.

Researchers in the past have shown that detectors tend to focus on noise level
features generated by CNNs more easily when trained with real data [14, 27].
We hypothesize that the model trained on synthetic data learns more
interpretable or higher-level image features as the synthetic images which are
considered real for the classification model already contain some CNN-
generated artifacts. These artifacts get overlayed on the ones introduced by
the face-swap algorithm making it difficult to distinguish between them.

##  6 Conclusion

In this paper, we present an approach for detecting face swaps using synthetic
data. We show that this dataless and privacy-aware approach of using faces
that do not exist achieves competitive performance to models trained on real
data. With some additional fine-tuning on real data, the model trained only on
synthetic data can actually surpass the performance of the model trained on
real data. In addition to the privacy and memory advantages of using synthetic
datasets, we show that the final trained model is also less biased and learns
more interpretable features. We measure the bias based on the performance of
the model across the different races, genders, and age groups. Overall, we
have shown several advantages to shifting from using real data for training
such models to using synthetic data.

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# Integral preserving discretization of 2D Toda lattices

Sergey V. Smirnov Department of Mathematics and Mechanics, Moscow State
University. E-mail<EMAIL_ADDRESS>
###### Abstract

There are different methods of discretizing integrable systems. We consider
semi-discrete analog of two-dimensional Toda lattices associated to the Cartan
matrices of simple Lie algebras that was proposed by Habibullin in 2011. This
discretization is based on the notion of Darboux integrability. Generalized
Toda lattices are known to be Darboux integrable in the continuous case (that
is, they admit complete families of characteristic integrals in both
directions). We prove that semi-discrete analogs of Toda lattices associated
to the Cartan matrices of all simple Lie algebras are Darboux integrable. By
examining the properties of Habibullin’s discretization we show that if a
function is a characteristic integral for a generalized Toda lattice in the
continuous case, then the same function is a characteristic integral in the
semi-discrete case as well. We consider characteristic algebras of such
integral-preserving discretizations of Toda lattices to prove the existence of
complete families of characteristic integrals in the second direction.

##  1 Introduction

Two-dimensional Toda lattice

$q_{i,xy}=\exp(q_{i}-q_{i+1})-\exp(q_{i-1}-q_{i})$ (1)

plays an important role both in classical differential geometry and in
mathematical physics. It is known to be Darboux integrable (i.e. it admits
complete family of essentially independent characteristic integrals), if the
trivial boundary conditions $q_{-1}=-\infty$ and $q_{r+1}=+\infty$ are imposed
for some natural $r$. Toda lattice can be rewritten in the form

$\displaystyle\left\\{\begin{array}[]{l}u^{1}_{xy}=\exp(2u^{1}-u^{2})\\\
u^{i}_{xy}=\exp(-u^{i-1}+2u^{i}-u^{i+1}),\quad i=2,\dots,r-1\\\
u^{r}_{xy}=\exp(-u^{r-1}+2u^{r})\end{array}\right.,$ (5)

where $q_{i}=u^{i+1}-u^{i}$. This system is a particular case of the so-called
exponential systems

$u^{i}_{xy}=\exp\left(\sum\limits_{j=1}^{r}a_{ij}u^{j}\right),\quad
i=1,2,\dots,r,$ (6)

that were introduced in [1] (here $a_{ij}$ are constant coefficients). Such
integrable generalizations of the Toda system corresponding to the Cartan
matrices $M=(a_{ij})$ of all simple Lie algebras were studied in a number of
papers in the beginning of 1980-ies [1]-[4]. Almost at the same time, discrete
versions of some of these systems started to appear in literature within the
frame of discretizing the theory of integrable systems [5, 6]. Due to the fact
that generalized Toda systems are related to many relevant mathematical
theories, the number of papers on the subject is enormous: continuous Toda
systems and their discretizations are addressed from different perspectives
and studied using various methods. Not even attempting to make a review, we
will mention the papers [7]-[17] that are important for our approach to the
problem of discretizeing Toda lattices.

Purely discrete versions of the generalized Toda lattice, associated with the
$A$-series Cartan matrices, were studied in [7, 8, 9]. Purely discrete and
semi-discrete versions of the $C$-series Toda lattice were examined in [11,
15]. Discretizations of generalized Toda lattices introduced in these papers
are based mostly on the notion of the Laplace invariants of hyperbolic
difference or differential-difference operators and on their specific
properties, and therefore this approach is not applicable in the general case
of an exponential system associated to arbitrary Cartan matrix.

There are several general approaches that lead to (semi)-discrete versions of
exponential systems (6) corresponding to Cartan matrices. Hamiltonian approach
was used in [10] to obtain semi-discrete analogs of exponential systems
associated to Cartan matrices of all simple Lie algebras. Another approach to
the problem of discretization of exponential systems was proposed in [13, 14].
The idea is to look for a semi-discrete system for the functions $u^{i}_{n}$,
depending on continuous variable $x$ and discrete variable $n$, such that its
characteristic $n$-integrals are given by the same formulas as $y$-integrals
of the continuous model. We will call this method integral preserving
discretization. This approach had appeared to be fruitful earlier in the study
of Darboux integrable semi-discrete scalar equations [18], before it was
applied to the Toda lattices. The analysis of the case $r=2$ from the
viewpoint of integral preserving method allowed the authors [13] to propose
the following discretization for exponential system (6) with the coefficient
matrix $M=(a_{ij})$:

$u^{i}_{n+1,x}-u^{i}_{n,x}=\exp\left(\sum\limits_{j=1}^{i-1}a_{ij}u^{j}_{n}+\frac{a_{ii}}{2}(u^{i}_{n}+u^{i}_{n+1})+\sum\limits_{j=i+1}^{r}a_{ij}u^{j}_{n+1}\right).$
(7)

Similar approach [14] allows to define the analogs of exponential systems in
the purely discrete case:

$\exp(u^{i}_{n+1,m+1}+u^{i}_{n,m}-u^{i}_{n+1,m}-u^{i}_{n,m+1})=\\\
=1+\exp\left(\sum\limits_{j=1}^{i-1}a_{ij}u^{j}_{n,m+1}+\frac{a_{ii}}{2}(u^{j}_{n,m+1}+u^{j}_{n+1,m})+\sum\limits_{j=i+1}^{r}a_{ij}u^{j}_{n+1,m}\right).$
(8)

In both cases, Darboux integrability was proved only for discretized systems
associated with all Cartan matrices of the rank $2$ [13, 14]. In the general
case, Darboux integrability for (semi)-discrete versions of the Toda lattice
is proven only for the $A$\- and $C$-series lattice by constructing a
generating function for characteristic integrals [19], but this method is not
applicable for the $B$-, $D$-series lattices and for the systems corresponding
to exceptional Cartan matrices $E_{6}$–$E_{8}$ and $F_{4}$.

Although discterizations of $A$-series Toda lattice given in [10] and in [13]
are the same, the methods used there produce different semi-discrete systems
for other series of Cartan matrices (continuum limits in both cases are the
same). Another version of semi-discrete Toda system of the series $B$ is
obtained in [16] by considering a modification of skew-orthogonal polynomials
that arise in the random matrix theory. In [17] direct linearization method
was used to study semi-discrete analogs of Toda systems corresponding to some
series of affine Cartan matrices. In the purely discrete case there also exist
different versions of Toda systems corresponding to Cartan matrices, see [14,
12].

Various properties of integrable models are usually taken as a basis for
finding discretizations. Discretization of Toda systems proposed in [13] is
based on the notion of Darboux integrability and hence it is important to show
that systems introduced in [13] are Darboux integrable indeed. In this paper
we focus on semi-discrete exponential systems (7) and prove that such systems
corresponding to the Cartan matrices of all simple Lie algebras are Darboux
integrable. Therefore we justify the integral preserving discretization method
for generalized Toda lattices. More precisely, we prove that if exponential
system (6) admits $y$-integral

$I=I(u^{1}_{x},\dots,u^{r}_{x},u^{1}_{xx},\dots,u^{r}_{xx},u^{1}_{xxx},\dots,u^{r}_{xxx},\dots),$

then the same function

$I_{n}=I(u^{1}_{n,x},\dots,u^{r}_{n,x},u^{1}_{n,xx},\dots,u^{r}_{n,xx},u^{1}_{n,xxx},\dots,u^{r}_{n,xxx},\dots),$

whose arguments are replaced by the dynamical variables for the semi-discrete
case, is an $n$-integral for discretization (7) of this system. Besides this,
using characteristic algebras, we prove that semi-discrete versions (7) of
$B$-, $D$-series Toda lattices and the systems associated to exceptional
Cartan matrices $E_{6}$–$E_{8}$ and $F_{4}$ admit complete families of
essentially independent characteristic $x$-integrals (Darboux integrability of
$A$\- and $C$-series lattices have been proved earlier). Altogether, this
proves Darboux integrability of all semi-discrete Toda lattices (7)
corresponding to the Cartan matrices of simple Lie algebras, which was
conjectured in [13].

The paper is structured as follows: in Section 2 we review the notions of
Darboux integrability, characteristic algebra and the relation between them.
In Section 3 we describe Habibullin’s method and prove the existence of a
complete family of characteristic $n$-integrals for semi-discrete lattices (7)
corresponding to the Cartan matrices of all simple Lie algebras and hence we
justify Habibullin’s integral preserving discretization method by showing that
if a function is a $y$-integral of discrete exponential system, then the same
function defines an $n$-integral for its semi-discrete analog. Basic
properties of characteristic algebras for exponential systems associated to
the Cartan matrices of simple Lie algebras are discussed in Section 4. In
Section 5 we prove the existence of a complete family of independent
$x$-integrals for semi-discrete exponential systems corresponding to the
Cartan matrices of all simple Lie algebras.

##  2 Darboux integrability and characteristic algebras

In the theory of integrable systems there are several different approaches to
integrability depending in the class in the systems that are being considered:
Liouville integrability, existence of a Lax pair, existence of higher
symmetries. Darboux integrability is a kind of “very strong” integrability
that is defined for hyperbolic systems and that is closely related to explicit
integrability. We start this Section with a series of definitions and notation
[1, 20] that will be used in this paper.

Function

$I=I(u^{1},\dots,u^{r},u^{1}_{x},\dots,u^{r}_{x},u^{1}_{xx},\dots,u^{r}_{xx},u^{1}_{xxx},\dots,u^{r}_{xxx},\dots)$

is called a $y$-integral of hyperbolic system

$u^{i}_{xy}=F^{i}(x,y,u^{1},\dots,u^{r},u^{1}_{x},\dots,u^{r}_{x},u^{1}_{y},\dots,u^{r}_{y}),\quad
i=1,\dots,r,$ (9)

if its total derivative with respect to $y$ by virtue of the system vanishes:

$0=D_{y}(I)=\sum\limits_{i=1}^{r}\left(u^{i}_{y}\frac{\partial I}{\partial
u^{i}}+F^{i}\frac{\partial I}{\partial u^{i}_{x}}+D_{x}(F^{i})\frac{\partial
I}{\partial u^{i}_{xx}}+D^{2}_{x}(F^{i})\frac{\partial I}{\partial
u^{i}_{xxx}}+\dots\right),$

where $D_{x}$ is the total derivative with respect to $x$. The highest order
of $x$-derivative of the functions $u^{1},\dots u^{r}$, on which $y$-integral
$I$ depends, is called the order of $I$; $y$-integral is called non-trivial if
it depends not only on the independent variable $x$. Here and further we will
consider only non-trivial integrals; $x$-integrals of hyperbolic system (9)
are defined similarly. Both $x$\- and $y$-integrals are called characteristic
integrals. Denote

$u^{i}_{1}=u^{i}_{x},\quad u^{i}_{2}=u^{i}_{xx},\quad
u^{i}_{3}=u^{i}_{xxx},\dots,\quad i=1,\dots,r.$

Characteristic $y$-integrals $I_{1},\dots,I_{k}$ of orders $d_{1},\dots d_{k}$
are called essentially independent if the rank of the matrix

$\displaystyle\left(\begin{array}[]{cccc}\frac{\partial I_{1}}{\partial
u^{1}_{d_{1}}}&\frac{\partial I_{1}}{\partial
u^{2}_{d_{1}}}&\dots&\frac{\partial I_{1}}{\partial u^{r}_{d_{1}}}\\\
\frac{\partial I_{2}}{\partial u^{1}_{d_{2}}}&\frac{\partial I_{2}}{\partial
u^{2}_{d_{2}}}&\dots&\frac{\partial I_{2}}{\partial u^{r}_{d_{2}}}\\\
\vdots&&\ddots&\\\ \frac{\partial I_{k}}{\partial
u^{1}_{d_{k}}}&\frac{\partial I_{k}}{\partial
u^{2}_{d_{k}}}&\dots&\frac{\partial I_{k}}{\partial u^{r}_{d_{k}}}\\\
\end{array}\right)$ (14)

is equal to $k$.

###### Definition 1

Hyperbolic system (9) is called Darboux integrable if it admits complete
families of essentially independent $x$\- and $y$-integrals.

###### Example 1

Liouville equation

$u_{xy}=\exp u$ (15)

is Darboux integrable since it admits characteristic integrals in both
directions: functions

$I=u_{xx}-\frac{1}{2}u_{x}^{2}\quad\hbox{and}\quad
J=u_{yy}-\frac{1}{2}u_{y}^{2}$ (16)

are $y$\- and $x$-integrals respectively.

Exponential systems associated with the Cartan matrices of simple Lie algebras
(i.e. exponential systems (6) such that the coefficient matrix is the Cartan
matrix of one of simple Lie algebras) are known to be Darboux integrable [2].
These systems are also known in literature as generalized Toda lattices
corresponding to the Cartan matrices of simple Lie algebras. Explicit formulas
for characteristic integrals in terms of wronskians were found in [21] for
Toda lattices of series $A$–$D$. Another approach that allows to obtain
generating function for characteristic integrals was developed in [22] for the
$A$-series Toda lattices and in [19] for lattices of the series $A$–$C$.

Characteristic integrals are two-dimensional analogs of first integrals for
ODEs, but there is an essential difference between these two cases: hyperbolic
equations having characteristic integrals are exceptional. If function $I$ is
a $y$-integral of (9), then its $x$-derivatives $D_{x}(I),D^{2}_{x}(I),\dots$
are obviously also $y$-integrals, but these integrals are not essentially
independent. The existence of a complete family of characteristic integrals is
controlled by an algebraic tool — Lie algebra of differential operators that
is called the characteristic algebra of hyperbolic system [23, 1, 4].
Characteristic algebra can be defined for arbitrary hyperbolic system of form
(9), but in this general case it should be considered as a Lie-Rinehart
algebra (see discussion in [24]). In the special case of exponential systems
(6) the situation is more simple and the characteristic algebra can be
referred to as a Lie algebra generated by differential operators of a certain
kind.

###### Definition 2

Lie algebra generated by operators

$\frac{\partial}{\partial u^{1}},\dots,\frac{\partial}{\partial u^{r}},\quad
D_{y}=\sum\limits_{i=1}^{r}\left(e^{w^{i}}\frac{\partial}{\partial
u^{i}_{1}}+D_{x}\left(e^{w^{i}}\right)\frac{\partial}{\partial
u^{i}_{2}}+D^{2}_{x}\left(e^{w^{i}}\right)\frac{\partial}{\partial
u^{i}_{3}}+\dots\right),$

where $w^{i}=a_{i1}u^{1}+\dots+a_{ir}u^{r}$, is called the characteristic
algebra of exponential system (6).

One can easily show that $y$-integrals of exponential systems (6) cannot
depend on $u^{1},\dots u^{r}$: they depend only on their $x$-derivatives.
Therefore any $y$-integral annihilates the whole characteristic algebra.

###### Remark 1

In the general case of hyperbolic systems (9), one has to define
characteristic algebra in the direction of the variable $x$ and characteristic
algebra in the direction $y$. For exponential systems (6), these Lie algebras
are isomorphic since variables $x$ and $y$ enter the equations symmetrically.

###### Proposition 1

Let the matrix $M=(a_{ij})$ of exponential system (6) be non-degenerate. Then
its characteristic algebra is generated by operators

$\frac{\partial}{\partial
u^{i}},\quad\tilde{X}_{i}=e^{w^{i}}\left(\frac{\partial}{\partial
u^{i}_{1}}+b^{i}_{1}\frac{\partial}{\partial
u^{i}_{2}}+b^{i}_{2}\frac{\partial}{\partial u^{i}_{3}}+\dots\right),\quad
i=1,\dots,r,$

where $b^{i}_{k}=b^{i}_{k}(w^{i}_{1},w^{i}_{2},\dots
w^{i}_{k})=e^{-w^{i}}D^{k}_{x}(e^{w^{i}})$ is the $k$-th complete Bell
polynomial of the variables $w^{i}_{1},w^{i}_{2},\dots w^{i}_{k}$ and
$w^{i}_{k}=D^{k}_{x}(w^{i})$.

Proof. Simple calculation shows that for all $i=1,\dots,r$

$\left[\frac{\partial}{\partial
u^{i}},D_{y}\right]=a_{i1}\tilde{X}_{1}+a_{i2}\tilde{X}_{2}+\dots+a_{ir}\tilde{X}_{r}.$

Hence, all operators of the form
$a_{i1}\tilde{X}_{1}+a_{i2}\tilde{X}_{2}+\dots+a_{ir}\tilde{X}_{r}$ belong to
the characteristic algebra, and it follows from non-degeneracy of the matrix
$M$ that operators $\tilde{X}_{i}$ are linear combinations of these operators.
Therefore, they belong to characteristic algebra, and since
$D_{y}=\tilde{X}_{1}+\dots+\tilde{X}_{r}$, they generate the characteristic
algebra together with $\frac{\partial}{\partial u^{i}}$, where $i=1,\dots,r$.
$\Box$

###### Theorem 1

[1] Exponential system (6) is Darboux integrable if and only if its
characteristic algebra is finite-dimensional.

###### Example 2

Characteristic algebra of the Liouville equation (15) is two-dimensional:
$\left[\frac{\partial}{\partial u},D_{y}\right]=D_{y}$.

###### Remark 2

Since

$\left[\frac{\partial}{\partial
u^{j}},\tilde{X}_{i}\right]=a_{ij}\tilde{X}_{i}$

for all $i,j=1,\dots,r$ and characteristic $y$-integrals of an exponential
system (6) cannot depend on $u^{1},\dots,u^{r}$, it is sufficient for the
study of Darboux integrability to consider reduced characteristic algebra
generated by $\tilde{X}_{1},\dots,\tilde{X}_{r}$: obviously, exponential
system (6) is Darboux integrable if and only if its reduced characteristic
algebra is finite-dimensional. Note that this Lie algebra is isomorphic to Lie
algebra generated by vector fields $X_{1},\dots X_{r}$ where
$X_{i}=e^{-w^{i}}\tilde{X}_{i}$ for all $i=1,\dots r$.

The notion of Darboux integrability can be extended to the case of
(semi)-discrete hyperbolic systems. Function

$I_{n}=I(u_{n}^{1},\dots,u_{n}^{r},u^{1}_{n,x},\dots,u^{r}_{n,x},u^{1}_{n,xx},\dots,u^{r}_{n,xx},u^{1}_{n,xxx},\dots,u^{r}_{n,xxx},\dots)$

is called an $n$-integral of semi-discrete hyperbolic system

$u^{i}_{n+1,x}-u^{i}_{n,x}=F^{i}(x,n,u^{1}_{n},\dots,u^{r}_{n},u^{1}_{n,x},\dots,u^{r}_{n,x},u^{1}_{n+1},\dots,u^{r}_{n+1}),\quad
i=1,\dots,r,$ (17)

if its total difference derivative by virtue of the system vanishes:
$I_{n+1}-I_{n}=0$. One can verify that in the semi-discrete case $n$-integrals
cannot depend on shifted variables $u^{i}_{n+1}$ and $x$-integrals cannot
depend on the derivatives $u^{i}_{n,x}$, where $i=1,\dots,r$. The order $d$ of
an $x$-integral is defined as the highest shift $u_{n+d}$ on which it depends.
Family of $x$-integrals $J_{1},\dots J_{k}$ of orders $d_{1},\dots d_{k}$ are
called essentially independent if the rank of the matrix

$\displaystyle\left(\begin{array}[]{cccc}\frac{\partial J_{1}}{\partial
u^{1}_{n+d_{1}}}&\frac{\partial J_{1}}{\partial
u^{2}_{n+d_{1}}}&\dots&\frac{\partial J_{1}}{\partial u^{r}_{n+d_{1}}}\\\
\frac{\partial J_{2}}{\partial u^{1}_{n+d_{2}}}&\frac{\partial J_{2}}{\partial
u^{2}_{n+d_{2}}}&\dots&\frac{\partial J_{2}}{\partial u^{r}_{n+d_{2}}}\\\
\vdots&&\ddots&\\\ \frac{\partial J_{k}}{\partial
u^{1}_{n+d_{k}}}&\frac{\partial J_{k}}{\partial
u^{2}_{n+d_{k}}}&\dots&\frac{\partial J_{k}}{\partial u^{r}_{n+d_{k}}}\\\
\end{array}\right)$ (22)

is equal to $k$. Similarly to the continuous case, hyperbolic system (17) is
called Darboux integrable if it admits complete families of essentially
independent $n$\- and $x$\- integrals. In the entirely discrete case, Darboux
integrability of hyperbolic system

$u^{i}_{n+1,m+1}-u^{i}_{n+1,m}-u^{i}_{n,m+1}+u^{i}_{n,m}=F^{i}(n,m,u^{1}_{n,m},\dots,u^{r}_{n,m},u^{1}_{n+1,m},\dots,u^{r}_{n+1,m},u^{1}_{n,m+1},\dots,u^{r}_{n,m+1}),$

where $i=1,\dots,r$, requires the existence of essentially independent
families of $m$\- and $n$-integrals.

Darboux integrability of semi-discrete and entirely discrete exponential
systems corresponding to the Cartan matrices of the series $A$ and $C$ (i.e.
generalized (semi)-discrete Toda lattices of the series $A$ and $C$) was
proved in [19]. Another approach allowing to obtain a complete family of
essentially independent $x$-integrals for semi-discrete $A$-series Toda
lattice in terms of casoratians was developed in [25].

##  3 Integral preserving discretization

There are many different ways to discretize integrable systems. One of the
popular methods to discretize a PDE with two independent variables is to
consider iterations of its Bäcklund transformations as shinfts in a new
discrete variable. Then the formula for Bäcklund transformation that relates
the unknown function $u$ with its Bäcklund-image $u_{1}$ is a differential-
difference equation and it can be referred to as a semi-discrete analog of the
initial PDE. In this case, the superposition formula plays the role of
entirely discrete analog. Although this approach is widely used in the theory
of integrable systems, we will use another approach proposed by Habibullin et
al. [18] that is based on the existence of characteristic integrals for
generalized Toda lattices, which is a very specific property for this class of
systems.

In this section, we describe the integral preserving discretization method for
exponential systems associated to the Cartan matrices of simple Lie algebras
and justify it by proving that this approach provides Darboux integrable semi-
discrete systems.

Goursat [26] had found a complete list of scalar hyperbolic equations having
both characteristic integrals of order not greater than $2$ (the so-called
Goursat list) within the frame of the study of hyperbolic equations that admit
characteristic integrals. The idea proposed in [18] is to take characteristic
$y$-integral for an equation in the Goursat list and to find semi-discrete
hyperbolic equation such that this function is its $n$-integral. The
discretization obtained using this method inherits the main property of its
continuous counterpart: by construction, it admits an $n$-integral of order
not greater than $2$. Surprisingly, all semi-discrete equations found in [18]
appear to be Darboux integrable, i.e. in addition to $n$-integrals they also
admit characteristic $x$-integrals. Moreover, this method was also applied to
these semi-discrete equations in order to get entirely discrete analogs of the
equations from the Goursat list [18].

The same method was used in [13] to guess an appropriate formula for semi-
discrete analog of exponential systems (6) corresponding to the Cartan
matrices of simple Lie algebras. More precisely, careful analysis of the
systems corresponding to the Cartan matrices of the rank $2$ and their
characteristic integrals allowed the authors to find semi-discrete analogs for
exponential systems of the rank $2$, to obtain formula (7) and to conjecture
that systems (7) are Darboux integrable for the Cartan matrices of all simple
Lie algebras. The following theorem is the main result of this section and it
proves the first part of the conjecture from [13] (that Habilullin’s
discretization preserves $y$-integrals); the second part that states the
existence of sufficient number of essentially independent $x$-integrals is
proved in section 5.

###### Theorem 2

Let
$I=I(u^{1}_{x},\dots,u^{r}_{x},u^{1}_{xx},\dots,u^{r}_{xx},u^{1}_{xxx},\dots,u^{r}_{xxx},\dots)$
be a $y$-integral of exponential system (6) with non-degenerate matrix
$M=(a_{ij})$. Then the same function

$I_{n}=I(u^{1}_{n,x},\dots,u^{r}_{n,x},u^{1}_{n,xx},\dots,u^{r}_{n,xx},u^{1}_{n,xxx},\dots,u^{r}_{n,xxx},\dots)$

is an $n$-integral of discretization (7) of this system.

Proof.

Let $I$ be a $y$-integral or order $d$ of the continuous system (6). Similarly
to the continuous case, denote $u^{i}_{n,k}=D_{x}^{k}(u^{i}_{n})$ and
$w^{i}_{n}=a_{i1}u^{i}_{n}+\dots+a_{ir}u^{i}_{r}$ for all $i=1,\dots,k$. Using
Taylor’s expansion, rewrite the difference $I_{n+1}-I_{n}$, where variables
$u^{i}_{n}$ satisfy semidiscrete system (7):

$\displaystyle I_{n+1}-I_{n}$
$\displaystyle=\sum_{i=1}^{r}\sum_{k=1}^{d}\frac{\partial I}{\partial
u^{i}_{k}}\left(u^{i}_{n+1,k}-u^{i}_{n,k}\right)+$
$\displaystyle+\frac{1}{2}\sum_{i_{1},i_{2}=1}^{r}\sum_{k_{1},k_{2}=1}^{d}\frac{\partial^{2}I}{\partial
u^{i_{1}}_{k_{1}}\partial
u^{i_{2}}_{k_{2}}}\left(u^{i_{2}}_{n+1,k_{2}}-u^{i_{2}}_{n,k_{2}}\right)\left(u^{i_{1}}_{n+1,k_{1}}-u^{i_{1}}_{n,k_{1}}\right)+\dots=$
$\displaystyle=\sum\limits_{s=1}^{\infty}\left(\frac{1}{s!}\sum_{\lambda_{ik}:\,\sum\limits_{i=1}^{r}\sum\limits_{k=1}^{d}\lambda_{ik}=s}\frac{s!}{\prod\limits_{i=1}^{r}\prod\limits_{k=1}^{d}\lambda_{ik}!}\prod\limits_{i=1}^{r}\prod\limits_{k=1}^{d}\left(u^{i}_{n+1,k}-u^{i}_{n,k}\right)^{\lambda_{ik}}\frac{\partial^{s}I}{(\partial
u^{1}_{1})^{\lambda_{11}}\dots(\partial u^{r}_{d})^{\lambda_{rd}}}\right)$
(23)

Denote

$E_{i}=\exp(w^{i}_{n}),\quad\Delta_{i}=\exp\left(\frac{a_{ii}}{2}\left(u^{i}_{n+1}-u^{i}_{n}\right)+a_{i,i+1}\left(u^{i+1}_{n+1}-u^{i+1}_{n}\right)+\dots+a_{ir}\left(u^{r}_{n+1}-u^{r}_{n}\right)\right),$

$t_{i}=E_{i}\Delta_{i},$ where $i=1,\dots,r$ (here we omit the dependence on
$n$ for simplicity in notation). Hence, the $i$-th equation of (7) can be
rewritten as $u^{i}_{n+1,x}-u^{i}_{n,x}=t_{i}$ and therefore we need to
simplify the terms of the form $D_{x}^{k-1}(t_{i})$ in (23) using equations
(7).

The remaining part of the proof of this Theorem is divided into a series of
propositions on some specific properties of the the functions $t_{i}$ and the
operators

$X_{i}=e^{-w^{i}}\tilde{X}_{i}=\frac{\partial}{\partial
u^{i}_{1}}+b^{i}_{1}\frac{\partial}{\partial
u^{i}_{2}}+b^{i}_{2}\frac{\partial}{\partial u^{i}_{3}}+\dots$

###### Proposition 2

For all $i=1,\dots,r$ functions $t_{i}$ satisfy the relation

$D_{x}(t_{i})=t_{i}\left(b^{i}_{1}+\frac{a_{ii}}{2}t_{i}+a_{i,i+1}t_{i+1}+\dots+a_{ir}t_{r}\right).$
(24)

Proof. Differentiate $t_{i}$ and use relations

$D_{x}(E_{i})=b^{i}_{1}E_{i},\quad
D_{x}(\Delta_{i})=\frac{a_{ii}}{2}t_{i}+a_{i,i+1}t_{i+1}+\dots+a_{ir}t_{r}.\quad\Box$

###### Proposition 3

Operators $X_{j}$ satisfy the relation

$X_{j}(D_{x}+b^{i}_{1})=(D_{x}+b^{i}_{1}+b^{j}_{1})X_{j}+a_{ij}$

for all $i,j=1,\dots,r$.

Proof. Complete Bell polynomials $B_{k}=B_{k}(v_{1},\dots,v_{k})$ are known to
satisfy the relation

$D_{x}(B_{k})=B_{k+1}-B_{1}B_{k},\quad k=1,2,\dots$

Apply this relation to calculate the commutator:

$\displaystyle\left[X_{j},D_{x}+b^{i}_{1}\right]$
$\displaystyle=\left(\frac{\partial}{\partial
u^{j}_{1}}+b^{j}_{1}\frac{\partial}{\partial
u^{j}_{2}}+b^{j}_{2}\frac{\partial}{\partial
u^{j}_{3}}+\dots\right)\left(b^{i}_{1}+\sum_{l=1}^{r}\left(u^{l}_{2}\frac{\partial}{\partial
u^{l}_{1}}+u^{l}_{3}\frac{\partial}{\partial u^{l}_{2}}+\dots\right)\right)-$
$\displaystyle-\left(b^{i}_{1}+D_{x}\right)\left(\frac{\partial}{\partial
u^{j}_{1}}+b^{j}_{1}\frac{\partial}{\partial
u^{j}_{2}}+b^{j}_{2}\frac{\partial}{\partial u^{j}_{3}}+\dots\right)=$
$\displaystyle=X_{j}(b^{i}_{1})+X_{j}(u^{j}_{2})\frac{\partial}{\partial
u^{j}_{1}}+X_{j}(u^{j}_{3})\frac{\partial}{\partial u^{j}_{2}}+\dots-
D_{x}(b^{j}_{1})\frac{\partial}{\partial
u^{j}_{2}}-D_{x}(b^{j}_{2})\frac{\partial}{\partial u^{j}_{3}}-\dots=$
$\displaystyle=a_{ij}+b^{j}_{1}\frac{\partial}{\partial
u^{j}_{1}}+b^{j}_{2}\frac{\partial}{\partial
u^{j}_{2}}+\dots+(b^{j}_{1}b^{j}_{1}-b^{j}_{2})\frac{\partial}{\partial
u^{j}_{2}}+(b^{j}_{1}b^{j}_{2}-b^{j}_{3})\frac{\partial}{\partial
u^{j}_{3}}+\dots=$
$\displaystyle=a_{ij}+b^{j}_{1}\left(\frac{\partial}{\partial
u^{j}_{1}}+b^{j}_{1}\frac{\partial}{\partial
u^{j}_{2}}+b^{j}_{2}\frac{\partial}{\partial
u^{j}_{3}}+\dots\right)=a_{ij}+b^{j}_{1}X_{j}.\quad\Box$

###### Proposition 4

Operators $X_{j}$ satisfy the relation

$X_{j}^{k}(D_{x}+b^{i}_{1})=(D_{x}+b^{i}_{1}+kb^{j}_{1})X_{j}+\left(ka_{ij}+\frac{k(k-1)}{2}a_{jj}\right)X_{j}^{k-1}$

for all $i,j=1,\dots,r$ and for all natural $k$.

Proof. Use induction in $k$. The base of induction follows from Proposition 3.
$\Box$

###### Proposition 5

Let $k_{1},\dots,k_{s}$ be arbitrary natural numbers and assume $1\leqslant
j_{1}<j_{2}<\dots<j_{s}\leqslant r$. Then

$X^{k_{s}}_{j_{s}}X^{k_{s-1}}_{j_{s-1}}\dots
X^{k_{1}}_{j_{1}}\left(D_{x}+b^{i}_{1}\right)=\left(D_{x}+k_{s}b^{j_{s}}_{1}+\dots+k_{1}b^{j_{1}}_{1}+b^{i}_{1}\right)X^{k_{s}}_{j_{s}}X^{k_{s-1}}_{j_{s-1}}\dots
X^{k_{1}}_{j_{1}}+\\\
+k_{s}\left(a_{ij_{s}}+\frac{k_{s}-1}{2}a_{j_{s}j_{s}}+k_{s-1}a_{j_{s-1}j_{s}}+k_{s-2}a_{j_{s-2}j_{s}}+\dots+k_{1}a_{j_{1}j_{s}}\right)X^{k_{s}-1}_{j_{s}}X^{k_{s-1}}_{j_{s-1}}\dots
X^{k_{1}}_{j_{1}}+\\\
+k_{s-1}\left(a_{ij_{s-1}}+\frac{k_{s-1}-1}{2}a_{j_{s-1}j_{s-1}}+k_{s-2}a_{j_{s-2}j_{s-1}}+k_{s-3}a_{j_{s-3}j_{s-1}}+\dots+k_{1}a_{j_{1}j_{s-1}}\right)X^{k_{s}}_{j_{s}}X^{k_{s-1}-1}_{j_{s-1}}\dots
X^{k_{1}}_{j_{1}}+\dots{\\\
}+k_{3}\left(a_{ij_{3}}+\frac{k_{3}-1}{2}a_{j_{3}j_{3}}+k_{2}a_{j_{2}j_{3}}+k_{1}a_{j_{1}j_{3}}\right)X^{k_{s}}_{j_{s}}\dots
X^{k_{3}-1}_{j_{3}}X^{k_{2}}_{j_{2}}X^{k_{1}}_{j_{1}}+\\\
+k_{2}\left(a_{ij_{2}}+\frac{k_{2}-1}{2}a_{j_{2}j_{2}}+k_{1}a_{j_{1}j_{2}}\right)X^{k_{s}}_{j_{s}}\dots
X^{k_{2}-1}_{j_{2}}X^{k_{1}}_{j_{1}}+k_{1}\left(a_{ij_{1}}+\frac{k_{1}-1}{2}a_{j_{1}j_{1}}\right)X^{k_{s}}_{j_{s}}\dots
X^{k_{2}}_{j_{2}}X^{k_{1}-1}_{j_{1}}$ (25)

for any $i=1,\dots,r$.

Proof. Use induction in $s$. The base of induction follows from Proposition 4.
$\Box$

###### Proposition 6

Let $k\in\mathbb{N}$. Then $D_{x}^{k}(t_{i})$ is a polynomial in
$t_{i},\dots,t_{r}$ of the degree $k+1$,

$D^{k}_{x}(t_{i})=\sum\limits_{\varkappa=1}^{k+1}C^{k}_{k_{i}\dots
k_{r}}t^{k_{i}}_{i}t^{k_{i+1}}_{i+1}\dots t^{k_{r}}_{r},$ (26)

where the summation is taken over all partitions $\varkappa=k_{i}+\dots+k_{r}$
into a sum of non-negative whole numbers such that $k_{i}>0$. Coefficients are
given by

$C^{k}_{k_{i}\dots k_{r}}=\frac{1}{k_{i}!\dots
k_{r}!}X^{k_{r}}_{r}X^{k_{r-1}}_{r-1}\dots
X^{k_{i+1}}_{i+1}X^{k_{i}-1}_{i}(b^{k}_{i}).$ (27)

Proof. Use induction in $k$. The base of the induction follows from (24) since
$X_{j}(b_{1}^{i})=a_{ij}$. Assume the proposition holds for

$D_{x}(t_{i}),\quad D^{2}_{x}(t_{i}),\dots,D^{k}_{x}(t_{i})$

and differentiate (26) with respect to $x$. The total derivative of the
coefficient $C^{k}_{k_{i}\dots k_{s}}$ contributes only to the coefficient of
$t^{k_{i}}_{i}\dots t^{k_{r}}_{r}$ in the expansion of $D^{k+1}_{x}$. Due to
(24), the derivative of $t^{k_{i}}_{i}\dots t^{k_{r}}_{r}$ contributes to the
coefficient of $t^{k_{i}}_{i}\dots t^{k_{r}}_{r}$ and to all coefficients of
the terms of degree $k_{i}+\dots+k_{r}+1$ such that only one power differs
from $k_{i},\dots,k_{r}$ by one. First consider one of the leading
coefficients $C^{k+1}_{k_{i}\dots k_{r}}$ in $D^{k+1}_{x}(t_{i})$. It appears
as the result of differentiation of the terms

$t^{k_{i}-1}_{i}t^{k_{i+1}}_{i+1}\dots t^{k_{r}}_{r},\quad
t^{k_{i}}_{i}t^{k_{i+1}-1}_{i+1}\dots
t^{k_{r}}_{r},\dots,t^{k_{i}}_{i}t^{k_{i+1}}_{i+1}\dots t^{k_{r}-1}_{r},$

where the corresponding power $k_{s}-1$ is non-negative. Hence,

$\displaystyle C^{k+1}_{k_{i}k_{i+1}\dots k_{r}}$
$\displaystyle=C^{k}_{k_{i}-1,k_{i+1}\dots
k_{r}}(k_{i}-1)\frac{a_{ii}}{2}+C^{k}_{k_{i}k_{i+1}-1\dots
k_{r}}\left(k_{i}a_{i,i+1}+(k_{i+1}-1)\frac{a_{i+1,i+1}}{2}\right)+$
$\displaystyle+C^{k}_{k_{i}k_{i+1}k_{i+2}-1\dots
k_{r}}\left(k_{i}a_{i,i+2}+k_{i+1}a_{i+1,i+2}+(k_{i+2}-1)\frac{a_{i+2,i+2}}{2}\right)+\dots+$
$\displaystyle+C^{k}_{k_{i}k_{i+1}-1\dots
k_{r}-1}\left(k_{i}a_{i,r}+k_{i+1}a_{i+1,r}+k_{i+2}a_{i+2,r}+\dots+k_{r-1}a_{r-1,r}+(k_{r}-1)\frac{a_{rr}}{2}\right)=$
$\displaystyle=\frac{1}{k_{i}!\dots
k_{r}!}\left(k_{i}\frac{k_{i}-1}{2}a_{ii}X^{k_{r}}_{r}\dots
X^{k_{i+1}}_{i+1}X^{k_{i}-2}_{i}(b^{k}_{i})+\right.$
$\displaystyle+k_{i+1}\left(k_{i}a_{i,i+1}+\frac{k_{i+1}-1}{2}a_{i+1,i+1}\right)X^{k_{r}}_{r}\dots
X^{k_{i+1}-1}_{i+1}X^{k_{i}-1}_{i}(b^{k}_{i})+\dots+$
$\displaystyle+\left.k_{r}\left(k_{i}a_{i,r}+k_{i+1}a_{i+1,r}+k_{i+2}a_{i+2,r}+\dots+k_{r-1}a_{r-1,r}+\frac{k_{r}-1}{2}a_{rr}\right)X^{k_{r}-1}_{r}\dots
X^{k_{i+1}}_{i+1}X^{k_{i}-1}_{i}(b^{k}_{i})\right).$ (28)

Apply now formula (25):

$X^{k_{r}}_{r}X^{k_{r-1}}_{r-1}\dots
X^{k_{i+1}}_{i+1}X^{k_{i}-1}_{i}(b^{k+1}_{i})=X^{k_{r}}_{r}X^{k_{r-1}}_{r-1}\dots
X^{k_{i+1}}_{i+1}X^{k_{i}-1}_{i}(D_{x}+b^{i}_{1})(b^{k}_{i})=\\\
=k_{r}\left(a_{ir}+\frac{k_{r}-1}{2}a_{rr}+k_{r-1}a_{r-1,r}+\dots+(k_{i}-1)a_{ir}\right)X^{k_{r}-1}_{r}X^{k_{r-1}}_{r-1}\dots
X^{k_{i}-1}_{i}(b^{k}_{i})+\dots+\\\
+k_{i+1}\left(a_{i,i+1}+\frac{k_{i+1}-1}{2}a_{i+1,i+1}+(k_{i}-1)a_{i,i+1}\right)X^{k_{r}}_{r}\dots
X^{k_{i+1}-1}_{i+1}X^{k_{i}-1}_{i}(b^{k}_{i})+\\\
+k_{i}\left(a_{ii}+\frac{k_{i}-2}{2}a_{ii}\right)X^{k_{r}}_{r}\dots
X^{k_{i+1}}_{i+1}X^{k_{i}-2}_{i}(b^{k}_{i}).$ (29)

Here we used relation

$X^{k_{r}}_{r}X^{k_{r-1}}_{r-1}\dots
X^{k_{i+1}}_{i+1}X^{k_{i}-1}_{i}(b^{k}_{i})=0,$

which holds since $k_{i}+\dots+k_{r}=k+2$. Comparing formulas (28) and (29)
proves the claim for leading coefficients in the expansion for
$D^{k+1}_{x}(t_{i})$.

The formula for non-leading coefficients is proved similarly, although the
calculation is more nasty in this case since one has to take into account the
terms $b^{s}_{1}t^{k_{i}}_{i}t^{k_{i+1}}_{i+1}\dots t^{k_{r}}_{r}$, where
$s=i,\dots r$, that come from the differentiation of
$t^{k_{i}}_{i}t^{k_{i+1}}_{i+1}\dots t^{k_{r}}_{r}$, and the terms that appear
from the differentiation of the coefficient $C^{k}_{k_{i}\dots k_{s}}$ which
are non-zero if $\varkappa<k+1$. Note that formula (25) has to be used also to
simplify the coefficient that comes from differentiation of $C^{k}_{k_{i}\dots
k_{s}}$. $\Box$

###### Proposition 7

Polynomial $D_{x}^{k}(t_{i})$ is divisible by $t_{i}$ for all $i=1,\dots,r$
and $k\in\mathbb{N}$. The coefficient of $t_{i}$ in $D_{x}^{k}(t_{i})$ equals
$b^{k}_{i}$:

$D_{x}^{k}(t_{i})=t_{i}(b^{k}_{i}+\dots).$

Proof. The first claim follows from (24). The second claim immediately follows
from (27). $\Box$

###### Proposition 8

Let $I$ be an analytic $y$-integral of exponential system (6). Then

$I_{n+1}-I_{n}=\sum\limits_{m=1}^{\infty}\left(\sum\limits_{k_{1}+\dots+k_{r}=m}\frac{1}{k_{1}!\dots
k_{r}!}X^{k_{r}}_{r}X^{k_{r-1}}_{r-1}\dots X^{k_{1}}_{1}(I)t_{1}^{k_{1}}\dots
t^{k_{r}}_{r}\right),$ (30)

where the sum is taken over all partitions $m=k_{1}+\dots+k_{r}$ such that
$k_{1},\dots,k_{r}\geqslant 0$.

Proof. Use induction in $m$. If $m=1$, then we need to prove that the
coefficient of $t_{i}$ in the sum (23) equals $X_{i}(I)$ for all
$i=1,\dots,r$. Since the monomial $t_{i}$ is contained only in the expansions
of form (26) for $t_{i},t_{i,x},t_{i,xx},\dots$, the coefficient of $t_{i}$ in
(23) equals

$1\cdot\frac{\partial I}{\partial u^{i}_{1}}+C^{1}_{1,0,\dots,0}\frac{\partial
I}{\partial u^{i}_{2}}+C^{2}_{1,0,\dots,0}\frac{\partial I}{\partial
u^{i}_{3}}+\dots=\frac{\partial I}{\partial u^{i}_{1}}+b^{i}_{1}\frac{\partial
I}{\partial u^{i}_{2}}+b^{i}_{2}\frac{\partial I}{\partial
u^{i}_{3}}+\dots=X_{i}(I)$

due to Proposition 7.

Assume now the coefficient $B$ of $t_{l}^{k_{l}}\dots t_{j}^{k_{j}}$ in (23)
has the form

$\frac{1}{k_{l}!\dots k_{j}!}X^{k_{j}}_{j}X^{k_{j-1}}_{j-1}\dots
X^{k_{l}}_{l}(I)$

for all $l<j$ and $k_{l}+\dots+k_{j}\leqslant m$. According to (23), the
coefficient $B$ is obtained from the product

$\prod\limits_{i=1}^{r}\prod\limits_{k=1}^{d}\left(D^{k-1}_{x}(t_{i})\right)^{\lambda_{ik}}$

by extracting the coefficients of $t_{l}^{k_{l}}\dots t_{j}^{k_{j}}$ and
multiplying them by appropriate multiple derivatives of $I$. Hence,

$B=\sum\limits_{p}\left(\left(\mu_{p}\prod\limits_{i,k}C^{k}_{i,q_{l}\dots
q_{j}}\right)\frac{\partial^{s}I}{(\partial
u^{1}_{1})^{\lambda_{11}}\dots(\partial u^{r}_{d})^{\lambda_{rd}}}\right),$

where $C^{k}_{i,q_{l}\dots q_{j}}$ are the coefficients of the form (27) in
one of the expansions of $D^{k}_{x}(t_{i})$ in (23), $0\leqslant
q_{i}\leqslant k_{i}$ for all $i=l,\dots,j$ (here $q_{l}\dots q_{j}$ depend on
$i$ and $k$) and $\mu_{p}\in\mathbb{R}$ are products of binomial coefficients
which appear when expansions of $D^{k}_{x}(t_{i})$ are raised to powers
$\lambda_{i,k}$ in (23). We do not need to specify the set of indices over
which the sum and the products are taken — we will only need the general form
of $B$.

Consider the coefficient $\tilde{B}$ of $t_{l}^{k_{l}}\dots
t_{j}^{k_{j}}t_{j+1}$ in (23). It is the sum of the terms of two types: the
first one is the product of the coefficient of $t_{l}^{k_{l}}\dots
t_{j}^{k_{j}}$ in (23) by the coefficient of $t_{j+1}$ in expansions of
$D^{k}_{x}(t_{j+1})$ for all $k=0,1,\dots$, and the second one appears when
$t_{j+1}$ is additionally taken from one of the expansions of $D^{k}(t_{i})$,
where $i<j+1$. For the terms of the first type, the derivative of the function
$I$ in (23) is differentiated by $u^{j+1}_{k+1}$ since the whole expression is
multiplied by $D^{k}_{x}(t_{j+1})$. According to Proposition 7 the coefficient
of $t_{j+1}$ in $D^{k}_{x}(t_{j+1})$ equals $b^{j+1}_{k}$. Therefore, it
follows from the inductive assumption that the contribution of the terms of
the first type to $\tilde{B}$ has the form

$\frac{1}{k_{l}!\dots k_{j}!}\left(X^{k_{j}}_{j}X^{k_{j-1}}_{j-1}\dots
X^{k_{l}}_{l}\left(\frac{\partial I}{\partial
u^{j+1}_{1}}\right)+b^{j+1}_{1}X^{k_{j}}_{j}X^{k_{j-1}}_{j-1}\dots
X^{k_{l}}_{l}\left(\frac{\partial I}{\partial u^{j+1}_{2}}\right)+\right.\\\
+\left.b^{j+1}_{2}X^{k_{j}}_{j}X^{k_{j-1}}_{j-1}\dots
X^{k_{l}}_{l}\left(\frac{\partial I}{\partial
u^{j+1}_{3}}\right)+\dots\right).$ (31)

Examine now the contribution of the terms of the second type to $\tilde{B}$.
Since $t_{j+1}$ is additionally taken from one of the expansions of
$D^{k}(t_{i})$ in this case, the terms contributing to $\tilde{B}$ have the
form

$\left(\mu_{p}\prod\limits_{i,k}C^{k}_{i,q_{l}\dots
q_{j}q_{j+1}}\right)\frac{\partial^{s}I}{(\partial
u^{1}_{1})^{\lambda_{11}}\dots(\partial u^{r}_{d})^{\lambda_{rd}}},$

where $q_{j+1}=1$ for only one pair $(i,k)$ and is zero for all other pairs
$(i,k)$. Note that due to (27)

$C^{k}_{i,q_{l}\dots q_{j}q_{j+1}}=X_{j+1}\left(C^{k}_{i,q_{l}\dots
q_{j}}\right)$ (32)

for all $i=1,\dots,r$ and for all $k\in\mathbb{N}$.

Since $X^{k_{j}}_{j}X^{k_{j-1}}_{j-1}\dots X^{k_{l}}_{l}(I)$ is a linear
combination of various multiple derivatives of the function $I$ with certain
coefficients and $X_{j+1}$ is a linear first order operator, all the terms in
the expression for

$X_{j+1}\left(\frac{1}{k_{l}!\dots k_{j}!}X^{k_{j}}_{j}X^{k_{j-1}}_{j-1}\dots
X^{k_{l}}_{l}(I)\right)$

can be grouped into two families: the operator $X_{j+1}$ is applied to the
derivatives of $I$ in the first family and it is applied to the coefficients
in the second family. Clearly, it follows from (31,32) and the Leibniz rule
that the first family corresponds to the terms of the first type in
$\tilde{B}$ and the second family corresponds to the terms of the second type.
Therefore, $\tilde{B}=X_{j+1}(B)$.

The proof that the coefficient of $t_{l}^{k_{l}}\dots t_{j}^{k_{j}+1}$ in (23)
has the form

$\frac{1}{k_{l}!\dots k_{j-1}!(k_{j}+1)!}X^{k_{j}+1}_{j}X^{k_{j-1}}_{j-1}\dots
X^{k_{l}}_{l}(I)$

is conducted in the same way, but one has to take into account additional
coefficient $k_{j}+1$ that comes from multiplicity in this case. $\Box$

Proof of Theorem 2 immediately follows from (30) since every $y$-integral
annihilates the whole characteristic algebra and, in particular, it
annihilates the generators $\tilde{X}_{1},\dots\tilde{X}_{r}$. Since
$X_{i}=e^{-w^{i}}\tilde{X}_{i}$, it also annihilates operators $X_{1},\dots
X_{r}$. $\Box$

##  4 Characteristic algebra of exponential system

In this Section we review a number of basic properties of characteristic
algebras for exponential systems corresponding to the Cartan matrices of
simple Lie algebras in the continuous case.

###### Lemma 1

Let

$X=\sum\limits_{i=1}^{r}\sum\limits_{k=1}^{\infty}P^{i}_{k}\frac{\partial}{\partial
u^{i}_{k}},\quad
D=\sum\limits_{i=1}^{r}\sum\limits_{k=0}^{\infty}u^{i}_{k+1}\frac{\partial}{\partial
u^{i}_{k}},$

where
$P^{i}_{k}=P^{i}_{k}(u^{1},\dots,u^{r},u^{1}_{1},\dots,u^{r}_{1},u^{1}_{2},\dots,u^{r}_{2},\dots)$
and $u^{i}_{0}=u^{i}$. If $[D,X]=0$, then $X=0$.

This lemma and its various analogs are widely used for explicit description of
characteristic algebras. The proof is straightforward and trivial.

###### Proposition 9

[1] For any $1\leqslant i_{1},\dots,i_{k}\leqslant r$ operators $X_{i}$
satisfy following commutation relations:

$[D,[X_{i_{1}},[X_{i_{2}},\dots,[X_{i_{k-1}},X_{i_{k}}]\dots]]]=-(b^{i_{1}}_{1}+\dots+b^{i_{k}}_{1})[X_{i_{1}},[X_{i_{2}},\dots,[X_{i_{k-1}},X_{i_{k}}]\dots]]-\\\
-(a_{i_{2}i_{1}}+a_{i_{3}i_{1}}+\dots+a_{i_{k}i_{1}})[X_{i_{2}},[X_{i_{3}},\dots,[X_{i_{k-1}},X_{i_{k}}]\dots]]-\\\
-(a_{i_{3}i_{2}}+a_{i_{4}i_{2}}+\dots+a_{i_{k}i_{2}})[X_{i_{1}},[X_{i_{3}},\dots,[X_{i_{k-1}},X_{i_{k}}]\dots]]-\dots{\\\
}-(a_{i_{k-1}i_{k-2}}+a_{i_{k}i_{k-2}})[X_{i_{1}},\dots,[X_{i_{k-3}},[X_{i_{k-1}},X_{i_{k}}]]\dots]-\\\
-a_{i_{k}i_{k-1}}[X_{i_{1}},\dots,[X_{i_{k-3}},[X_{i_{k-2}},X_{i_{k}}]]\dots]+a_{i_{k-1}i_{k}}[X_{i_{1}},\dots,[X_{i_{k-3}},[X_{i_{k-2}},X_{i_{k-1}}]]\dots].$
(33)

Technical Propositions 10–14 can be proved straightforwardly using Lemma 1.

###### Proposition 10

Reduced characteristic algebra of exponential system (6) corresponding to the
$A$-series Cartan matrix of the rank $r$ is a linear span of the following
vector fields:

$\displaystyle X_{k},\quad k=1,2,\dots,r,$
$\displaystyle[X_{k},[X_{k+1},\dots,[X_{l-1},X_{l}]\dots]],\quad 1\leqslant
k<l\leqslant r.$

###### Proposition 11

Reduced characteristic algebra of exponential system (6) corresponding to the
$B$-series Cartan matrix of the rank $r$ is a linear span of the following
linearly independent vector fields:

$\displaystyle X_{k},\quad k=1,2,\dots,r,$
$\displaystyle[X_{k},[X_{k+1},\dots,[X_{l-1},X_{l}]\dots]],\quad 1\leqslant
k<l\leqslant r,$
$\displaystyle[X_{k},[X_{k-1},[X_{k-2},\dots,[X_{2},[X_{1},[X_{1},[X_{2},\dots,[X_{l-2},[X_{l-1},X_{l}]]\dots]]]]\dots]]],\quad
1\leqslant k<l\leqslant r.$

###### Proposition 12

Reduced characteristic algebra of exponential system (6) corresponding to the
$C$-series Cartan matrix of the rank $r$ is a linear span of the following
linearly independent vector fields:

$\displaystyle X_{k},\quad k=1,2,\dots,r,$
$\displaystyle[X_{k},[X_{k+1},\dots,[X_{l-1},X_{l}]\dots]],\quad 1\leqslant
k<l\leqslant r,$
$\displaystyle[X_{k},[X_{k-1},[X_{k-2},\dots,[X_{2},[X_{1},[X_{2},\dots,[X_{l-2},[X_{l-1},X_{l}]]\dots]]]\dots]]],\quad
2\leqslant k\leqslant l\leqslant r.$

###### Proposition 13

Reduced characteristic algebra of exponential system (6) corresponding to the
$D$-series Cartan matrix of the rank $r$ is a linear span of the following
linearly independent vector fields:

$\displaystyle X_{k},\quad k=1,2,\dots,r,$
$\displaystyle[X_{k},[X_{k+1},\dots,[X_{l-1},X_{l}]\dots]],\quad 2\leqslant
k<l\leqslant r,$
$\displaystyle[X_{1},[X_{2},\dots,[X_{l-1},X_{l}]\dots]],\quad 3\leqslant
l\leqslant r,$ $\displaystyle[X_{1},[X_{3},\dots,[X_{l-1},X_{l}]\dots]],\quad
3\leqslant l\leqslant r,$
$\displaystyle[X_{k},[X_{k-1},[X_{k-2},\dots,[X_{3},[X_{1},[X_{2},[X_{3},\dots,[X_{l-2},[X_{l-1},X_{l}]]\dots]]]]\dots]]],\quad
3\leqslant k<l\leqslant r.$

###### Remark 3

The above bases for the series $A$ and $D$ can be found in [1] but with no
proofs. Detailed proofs for all these four cases can be found in [35].

###### Remark 4

One can verify that in characteristic algebras of exponential systems
corresponding to the Cartan matrices of the series $A$–$D$ there are no non-
trivial relations between multiple commutators of the form
$[X_{i_{1}},[X_{i_{2}},\dots,X_{i_{k}}]]$. More precisely, any such non-zero
commutator is proportional to some element from the corresponding basis (see
Propositions 10–13).

###### Proposition 14

Reduced characteristic algebra of exponential systems (6) corresponding to the
Cartan matrices of the $E$-series, of $G_{2}$ and $F_{4}$ root systems are
finite-dimensional, and in each of these cases there exist a basis such that
any non-zero multiple commutator $[X_{i_{1}},[X_{i_{2}},\dots,X_{i_{k}}]]$ is
proportional to some element from this basis111We do not provide here suitable
bases because they cannot be represented in a compact form like in
Propositions 10–13..

##  5 Characteristic $x$-integrals of semi-discrete exponential systems

Integral preserving discretization (7) of a Darboux integrable exponential
system (6) admits a complete family of independent $n$-integrals. Darboux
integrability of a semi-discrete system also requires the existence of a
complete family of essentially independent $x$-integrals, and since the
variables $x$ and $n$ do not enter the equations symmetrically (one of them is
continuous and the other one is discrete), such family of integrals cannot be
obtained just by renaming the variables (as in the continuous case). Complete
family of characteristic integrals was obtained explicitly in [19] for semi-
discrete $A$\- and $C$-series lattices, but this approach is not applicable
for the lattices associated with the Cartan matrices of other simple Lie
algebras since it is not known whether they are reductions of an $A$-series
lattice or not. In this section we are going to prove the existence of
complete families of characteristic $x$-integrals for semi-discrete lattices
corresponding to the Cartan matrices of all simple Lie algebras using the
concept of characteristic algebra.

Characteristic algebras for (semi)-discrete hyperbolic system were defined in
[27]–[29]; these Lie algebras for (semi)-discrete exponential systems (7),(8)
associated with the Cartan matrices of rank $2$ where described explicitly in
terms of generators and relations in [13, 14]. Characteristic algebras are an
effective tool in the study of Darboux integrability of hyperbolic equations
both in the continuous and in the (semi)-discrete cases (see papers [30]–[34]
by Ufa mathematical school).

In this section we define a special Lie algebra of differential operators that
controls the existence of complete family of $x$-integrals for semi-discrete
exponential systems (7). Although this Lie algebra is very similar by its
properties to characteristic algebra defined in [28] and its construction is
based on the same ideas, these Lie algebras are not isomorphic.

We will not describe the construction of characteristic algebra in the semi-
discrete case from [28] here. Instead, we introduce a Lie algebra with similar
properties that allows to prove Darboux integrability of all exponential
systems associated with the Cartan matrices of simple Lie algebras.

Consider semi-discrete exponential system (7) and introduce the following
notation:

$v^{i}_{n}=u^{i}_{n+1}-u^{i}_{n},\quad
w^{i}_{n}=a_{i1}u^{1}_{n}+\dots+a_{ir}u^{r}_{n},\quad
z^{i}_{n}=w^{i}_{n+1}-w^{i}_{n},\quad\Delta^{i}_{n}=\exp\left(\frac{a_{ii}}{2}v^{i}_{n}+a_{i,i+1}v^{i+1}_{n}+\dots+a_{ir}v^{r}_{n}\right),$

where $i=1,\dots,r$. Hence, equations (7) can be rewritten as

$v^{i}_{n,x}=e^{w^{i}_{n}}\Delta^{i}_{n},\quad i=1,\dots,r.$ (34)

Consider differential operators

$Y_{i}=c^{i}_{0}\frac{\partial}{\partial
v^{i}_{n}}+c^{i}_{1}\frac{\partial}{\partial
v^{i}_{n+1}}+c^{i}_{2}\frac{\partial}{\partial v^{i}_{n+2}}+\dots,\quad
i=1,\dots,r,$

where

$c^{i}_{0}=\Delta^{i}_{n},\quad
c^{i}_{1}=\exp\left(z^{i}_{n}\right)\Delta^{i}_{n+1},\dots,c^{i}_{k}=\exp\left(z^{i}_{n}+z^{i}_{n+1}+\dots+z^{i}_{n+k-1}\right)\Delta^{i}_{n+k},\quad
i=1,\dots,r.$ (35)

###### Proposition 15

Let the matrix $M=(a_{ij})$ of exponential system (7) be non-degenerate. Then
function

$J=J(v^{1}_{n},\dots,v^{r}_{n},v^{1}_{n+1},\dots,v^{r}_{n+1},v^{1}_{n+2},\dots,v^{r}_{n+2},\dots)$

is an $x$-integral of (7) if and only if it annihilates operators
$Y_{1},\dots,Y_{r}$.

Proof. Since

$v^{i}_{n+1,x}=e^{w^{i}_{n+1}}\Delta^{i}_{n+1}=e^{w^{i}_{n}}\exp(z^{i}_{n})\Delta^{i}_{n+1}$

due to (34), one can easily verify by induction that

$v^{i}_{n+k,x}=e^{w^{i}_{n}}\exp(z^{i}_{n}+z^{i}_{n+1}+\dots+z^{i}_{n+k-1})\Delta^{i}_{n+k}.$

Hence, apply the total derivative with respect to $x$:

$D_{x}(J)=\sum\limits_{i=1}^{r}\sum\limits_{k=0}^{\infty}v^{i}_{n+k,x}\frac{\partial
J}{\partial
v^{i}_{n+k}}=\sum\limits_{i=1}^{r}\sum\limits_{k=0}^{\infty}e^{w^{i}_{n}}\exp(z^{i}_{n}+z^{i}_{n+1}+\dots+z^{i}_{n+k-1})\Delta^{i}_{n+k}\frac{\partial
J}{\partial v^{i}_{n+k}}=\\\
=\sum\limits_{i=1}^{r}\sum\limits_{k=0}^{\infty}e^{w^{i}_{n}}c^{i}_{k}\frac{\partial
J}{\partial v^{i}_{n+k}}=\sum\limits_{i=1}^{r}e^{w^{i}_{n}}Y_{i}(J).$

Therefore, if $M$ is non-degenerate, then exponents $e^{w^{i}_{n}}$ are
linearly independent and $J$ is an $x$-integral if and only if it annihilates
operators $Y_{1},\dots,Y_{r}$. $\Box$

Variables

$u^{1}_{n},\dots,u^{r}_{n},v^{1}_{n},\dots,v^{r}_{n},v^{1}_{n+1},\dots,v^{r}_{n+1},v^{1}_{n+2},\dots,v^{r}_{n+2},\dots$

are independent. Lie algebra $\tilde{\mathcal{L}}$ generated by vector fields

$\frac{\partial}{\partial u^{1}_{n}},\dots,\frac{\partial}{\partial
u^{r}_{n}},\ \tilde{Y}_{1},\dots,\tilde{Y}_{r},$

where $\tilde{Y}_{i}=e^{w^{i}_{n}}Y_{i}$ for all $i=1,\dots,r$, is a discrete
analog of the characteristic algebra for exponential system (6). Similarly,
Lie algebra $\mathcal{L}$ generated by vector fields $Y_{1},\dots,Y_{r}$ is a
discrete analog of the reduced characteristic algebra for (6). Nevertheless,
in order to avoid ambiguity in the use of terminology, we will call this Lie
algebra the defining algebra for semi-discrete system (7). The following
theorem is proved exactly as Theorem 1 in the continuous case.

###### Theorem 3

Semi-discrete exponential system (7) admits a complete family of essentially
independent $x$-integrals if and only if its defining algebra is finite-
dimensional.

We are going to prove the existence of a complete family of independent
$x$-integrals for semi-discrete exponential systems (7) corresponding to the
Cartan matrices of all simple Lie algebras by applying Theorem 3 and by
describing the defining algebras $\mathcal{L}$ explicitly.

###### Proposition 16

Let $T$ be the shift operator, $T(v_{n})=v_{n+1}$. Then

$T[Y_{i_{1}},[Y_{i_{2}},\dots,[Y_{i_{k-1}},Y_{i_{k}}]\dots]]T^{-1}=\exp(-(z^{i_{1}}_{n}+\dots+z^{i_{k}}_{n}))\left([Y_{i_{1}},[Y_{i_{2}},\dots,[Y_{i_{k-1}},Y_{i_{k}}]\dots]]-\right.\\\
-c^{i_{1}}_{n}(a_{i_{2}i_{1}}+a_{i_{3}i_{1}}+\dots+a_{i_{k}i_{1}})[Y_{i_{2}},[Y_{i_{3}},\dots,[Y_{i_{k-1}},Y_{i_{k}}]\dots]]-\\\
-c^{i_{2}}_{n}(a_{i_{3}i_{2}}+a_{i_{4}i_{2}}+\dots+a_{i_{k}i_{2}})[Y_{i_{1}},[Y_{i_{3}},\dots,[Y_{i_{k-1}},Y_{i_{k}}]\dots]]-\dots{\\\
}-c^{i_{k-2}}_{n}(a_{i_{k-1}i_{k-2}}+a_{i_{k}i_{k-2}})[Y_{i_{1}},\dots,[Y_{i_{k-3}},[Y_{i_{k-1}},Y_{i_{k}}]]\dots]-\\\
-c^{i_{k-1}}_{n}a_{i_{k}i_{k-1}}[Y_{i_{1}},\dots,[Y_{i_{k-3}},[Y_{i_{k-2}},Y_{i_{k}}]]\dots]\left.+c^{i_{k}}_{n}a_{i_{k-1}i_{k}}[Y_{i_{1}},\dots,[Y_{i_{k-3}},[Y_{i_{k-2}},Y_{i_{k-1}}]]\dots]\right)+\dots,$
(36)

where the dots in the end stand for multiple commutators of the degrees
$1,\dots,k-2$.

Proof

It follows from relations (35) that

$TY_{i}T^{-1}=e^{-z^{i}_{n}}Y_{i}$

for all $i=1,\dots,r$. Formula (36) is proved by induction using relation

$Y_{i_{0}}\left(\exp(-(z^{i_{1}}_{n}+\dots+z^{i_{k}}_{n}))\right)=-c^{i_{0}}_{n}(a_{i_{1}i_{0}}+a_{i_{2}i_{0}}+\dots+a_{i_{k}i_{0}})$

and representation

$T[Y_{i_{0}},[Y_{i_{1}},[Y_{i_{2}},\dots,[Y_{i_{k-1}},Y_{i_{k}}]\dots]]]T^{-1}=\\\
=\left(TY_{i_{0}}T^{-1}\right)\left(T[Y_{i_{1}},[Y_{i_{2}},\dots,[Y_{i_{k-1}},Y_{i_{k}}]\dots]]T^{-1}\right)-\left(T[Y_{i_{1}},[Y_{i_{2}},\dots,[Y_{i_{k-1}},Y_{i_{k}}]\dots]]T^{-1}\right)\left(TY_{i_{0}}T^{-1}\right).\quad\Box$

###### Theorem 4

The defining algebra of semi-discrete exponential system (7) corresponding to
the Cartan matrix of any simple Lie algebra is finite-dimensional.

Proof. We will prove that the defining algebra of semi-discrete exponential
system (7) corresponding to the Cartan matrix of any simple Lie algebra is
isomorphic to the reduced characteristic algebra of its continuous analog.
Since in characteristic algebras corresponding to the Cartan matrices of all
simple Lie algebras there are no relations between non-trivial multiple
commutators except for the skew-symmetry, the Jacobi identity and their
implications (see Propositions 10–14), it follows from Lemma 1 and from
formula (33) that commutator

$[X_{i_{1}},[X_{i_{2}},\dots,[X_{i_{k-1}},X_{i_{k}}]\dots]]$ (37)

is trivial if and only if all terms in (33) vanish. Therefore, either
commutator

$[X_{i_{1}},[X_{i_{2}},\dots,[X_{l-1},[X_{l+1},\dots,[X_{i_{k-1}},X_{i_{k}}]\dots]]\dots]]$

is trivial, or its coefficient

$a_{i_{l+1}i_{l}}+a_{i_{l+2}i_{l}}+\dots+a_{i_{k}i_{l}}$

is zero for all $l=1,\dots,k$.

For each of the Cartan matrices the isomorphism is established by induction in
the degree of multiple commutators. Note that the coefficients of multiple
commutators of degree $k-1$ in formula (36) are the same as the coefficients
in (33). If commutator (37) is trivial, then all the terms in (33) vanish.
Hence, relation (36) for the corresponding multiple commutator of
$Y_{i_{1}},\dots,Y_{i_{k}}$ does not contain terms of degree $k-1$. Careful
analysis of the relations between coefficients in formula (36) for multiple
commutators of degrees $l$ and $l-1$ for each Cartan matrix shows that the
absence of the terms of degree $k-1$ yields the absence of all terms of
degrees $1,\dots,k-2$. Hence, commutator (37) satisfies relation

$T[Y_{i_{1}},[Y_{i_{2}},\dots,[Y_{i_{k-1}},Y_{i_{k}}]\dots]]T^{-1}=\exp(-(z^{i_{1}}_{n}+\dots+z^{i_{k}}_{n}))[Y_{i_{1}},[Y_{i_{2}},\dots,[Y_{i_{k-1}},Y_{i_{k}}]\dots]]$

which holds only if $k=1$. Therefore, if commutator (37) is trivial in the
reduced characteristic algebra of exponential system in the continuous case,
then the corresponding multiple commutator of $Y_{i_{1}},\dots,Y_{i_{k}}$ also
vanishes. This proves the isomorphism. $\Box$

###### Remark 5

We used the concept of characteristic algebra in order to prove the existence
of a complete family of $x$-integrals. Although our attempts to find explicit
formulas for $x$-integrals did not give any result in general, some integrals
can be found explicitly. For example, semi-discrete Toda lattice (7)
corresponding to the $B$-series Cartan matrix of the rank $r\geqslant 3$
admits $x$-integral

$J=\sum_{j=2}^{r-1}\left(\exp(v^{j+1}_{n+j+1}-v^{j}_{n+j+1})+\exp(v^{i}_{n}-v^{j+1}_{n+1})\right)+\exp(v^{2}_{n+1}-2v^{1}_{n+1})+\\\
+\exp(2v^{1}_{n}-v^{2}_{n+1})+2\exp(v^{1}_{n}-v^{1}_{n+1})+\exp(v^{r}_{n})+\exp(-v^{r}_{n+r}).$

###### Remark 6

It would have been interesting to prove that Habibullin’s integral preserving
discretization leads to Darboux integrable systems in the purely discrete case
as well. Entirely discrete exponential systems were introduced in [14] and it
was proved there that for all Cartan matrices of the rank $2$ characteristic
$x$-integrals of corresponding semi-discrete lattices appear to be
$m$-integrals of their purely discrete analogs. It was conjectured in [14]
that the same property holds for purely discrete exponential systems
associated with the Cartan matrices of all simple Lie algebras, but
unfortunately our approach appears to be not applicable for proving the
integral preserving property in the entirely discrete case due to the form of
equations (8).

##  6 Acknowledgements

The author wishes to thank Prof. Alexander Bobenko and TU Berlin for
hospitality. Research is partially funded by the Deutsche
Forschungsgemeinschaft (DFG) – TRR109 Collaborative Research Center
“Discretization in Geometry and Dynamics” (Project ID: 195170736).

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