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#Pyth, 64 48 50 bytes

#jdhfqzsTsm^+msXtjCk2U2"<>""
\r.1"",<..>,"dlzB

Test case.

Version that doesn't take forever ($O(9^{n/3}$) here, in 52 bytes.

This is the brute force approach, generate all sequences and check whether any sum to the input. Fish diagrams compressed as characters, whose binary representations are the > and <. The whole thing is wrapped in a try-catch block so that no output occurs when no results are found.

This is an $O(9^n)$ solution.

Some characters are stripped above, because control characters are used. They are reproduced faithfully at the link above.

xxd output:

0000000: 236a 6468 6671 7a73 5473 6d5e 2b6d 7358  #jdhfqzsTsm^+msX
0000010: 746a 436b 3255 3222 3c3e 2222 0a5c 7212  tjCk2U2"<>"".\r.
0000020: 141b 1d2e 3122 222c 3c2e 2e3e 2c22 646c  ....1"",<..>,"dl
0000030: 7a42                                     zB