You can still use asymmetric encryption: generate a public-private key pair, encrypt with the private key, and the client (the "application" as you mentioned) can decrypt it with the public key.
Of course, a public key (and the private key too) can be used for both encrypting and decrypting.
But re-encrypting plain text with a public key would generate a completely different cypher than a cypher encrypted using a private key. Decrypting and re-encrypting with the same key would produce a different result, and so it's useless.