Construct a musical piece that follows the designated lettered structure and integrates the provided motif throughout.
X:1
L:1/16
M:2/4
K:A
['agae agae', 'ceAe ceAe']  
X:1
L:1/16
M:2/4
K:A
"^Fast" e2 | agae agae | agae fedc | dfBf"^Rocking bow" dfBf dfBf agfe | agae agae | agae fedc | 
 dfed cABG | E2A2 A2>e2 ::"^Rocking bow" ceAe ceAe | ceAe fedc | dfBf dfBf | dfBf agfe | 
 ceAe ceAe | ceAe fedc | dfed cAB^G | E2A2 A,2 :|  