Comments on the Negative grade KdV Hierarchy

1. The report provides a detailed account of a contribution to the 34th International Colloquium on Group Theoretical Methods in Physics.

2. The authors use previously developed algebraic techniques and extend their (and other authors') earlier work to demonstrate a new interesting relationship between the KdV and mKdV hierarchies.

3. The relationship between the hierarchies and some of the consequences of the discovered relationship are described in detail.

1. The significance of the different 'vacuum structures' should be explained.

2. The article does need to be proof-read carefully because there are many grammatical errors, including incorrectly spelled words, missing definite and indefinite articles, missing punctuation, etc. I have supplied examples below.

The conference is devoted to the the many applications of group theory in physics, including mathematical physics, and the development of group-theoretical methods. This article falls within the scope of the meeting.

The article contains new results that should be of use to mathematical physicists interested in the structure of integrable field theories and their organisation within 'hierarchies'.

There is a limit of eight pages maximum for parallel session submissions to the conference and this article meets that restriction.

1. Equations

Eq(34) is stated to be an equation of motion. However, it is not actually an equation but an expression (there is no '=' anywhere).

Eq(49) is too long and needs to be reformatted to fit properly.

2. Spellings/insertions (this is already extensive but probably not exhaustive)

p1 l4 'which allows its' --> 'allowing their'
p2 l1 'properties as' --> 'properties, such as' and a ',' after 'solutions'
p2 l7 'negative [3]' --> 'negative flows [3]'
p2 l15 'there shall --> 'there should'
p3 insert ',' at the end of eq(8)
p3 insert ',' at the end of eq(14)
p3 'well know' --> 'well known'
p3 'be showed that' --> 'be shown that'
p3 insert ',' at the end of eq(16)
p3 to p4 Suggest placing '.' at the end of eq(18) and then breaking the long sentence into several short ones to include eqs(19,20) in separate sentences.
p4 before eq(23) replace 'Propose' --> 'We propose'
p4 after eq(23) insert 'the' before 'KdV' and also before 'sub-hierarchy'
p4 insert ',' at the ends of eqs(24,29)
p5 l1 and l4 'KdV' --> 'the KdV'
p5 l6 insert 'operator' after 'Lax'
p5 after eq(31) insert 'the' before 'inverse', replace 'recurssion' --> 'recursion', replace 'hamiltonian' --> 'Hamiltonian'
p5 before eq(32) insert 'operator' after 'Lax' and insert ',' at the end of eq(32)
p5 eq(34) seems to be missing something since it is evidently not an equation of motion
p5 after eq(36) insert 'whether' after 'no matter', replace 'are' by 'is' before 'associated'
p5 why is part of the final sentence italicised? (It might be better to split the sentence containing eqs(35,36) into two sentences. In any case, eq(36) should end with '.' and on the following line 'this' --> 'This')
p6 insert ',' at the end of eq(37)
p6 after eq(39) replace 'as local' --> 'as a local'
p7 l1-2 is this a new paragraph? If not, there is no need to start a new line with 'For'; if it is then there should be a proper paragraph break
p7 l2 replace 'satisfy' --> 'satisfies'
p7 insert ',' at the ends of eqs(45,46)
p7 replace 'Sinh' --> 'sinh' and insert 'the' before 'Miura'
p7 final paragraph: replace 'inherits' --> 'inherit', replace 'structure' --> 'structures', insert 'and' before 'each KdV', replace 'pair' --> 'pairs', remove space after 'multisolitons', remove 'its' before 'vacuum', change 'its' --> 'their', change 'in the lines' --> 'along the lines'
p8 ref[6] remove space and the extra ',' after the author's name