Rank 2 vector bundles in a neighborhood of an exceptional curve of a smooth surface

Let D congruent to P-1 be an exceptional divisor on the smooth surface W an d U the formal neighborhood of D in W. Let E be a rank 2 vector bundle on U . Here we associate to E an integer t greater than or equal to 1, a finite family E-i, 1 less than or equal to i less than or equal to t, of rank 2 ve ctor bundles on U and a finite sequence {(a(i), b(i))}1 less than or equal to i less than or equal to t of pairs of integers such that E-i\D has split ting type (a(i),b(i)), E-1 = E, a(t) = b(t), a(i + 1) + b(i + 1) = a(1) + b (1) + i and b(i) < b(i + 1) less than or equal to a(i + 1) less than or equ al to a(i) for 2 less than or equal to i less than or equal to t. Vice vers a, for any such sequence we prove the existence of at least one such bundle . We compute the second Chern class of E in terms of {(a(i),b(i))}1 less th an or equal to i less than or equal to t and show that O-U(-a(1)D)+ O-U(-b( 1)D) is the unique bundle with splitting type (al, bl) and maximal c(2).