Litecoinj transaction sent to unknown address
I looked into this by converting the addresses from base 58 to hex. What I got was this:
MD1AYTZnTtGKQToj2LTD7qLsSzrqcdXr9J -> 0x3237fb62589f848d3808f0b9bfe9762edabe150b93f409d315
LaPJ9TfBr6E2MSRa4t1MMhFPQAue4osUn8 -> 0x3037fb62589f848d3808f0b9bfe9762edabe150b93bab51bf3

Notice that the positions from byte 1 to byte 20 are the same: 37fb...0b93. 

This is unfortunately bad news and means that the software you are using is bad.

The address that starts with M or 3 in Litecoin are pay-to-script-hash addresses (P2SH) and the software should have created P2SH outputs but it has for some reason instead created pay-to-public-key-hash outputs (P2PKH) as you can see here: 

{
          "value": 15149,
          "script": "76a914a64e3a77893db78edfb3ba4abfa2106f84ba80eb88ac",
          "addresses": [
            "LaPJ9TfBr6E2MSRa4t1MMhFPQAue4osUn8"
          ],
          "script_type": "pay-to-pubkey-hash"
        },

The funds you have sent are unfortunately lost and you should stop using this software immediately since it only seems to support P2PKH addresses and not P2SH addresses.

If it is any comfort, you are not the first one to make this mistake.