Is there a way to do single key-pair asymmetric encryption?
Yes, RSA is an example of a cryptosystem where this is possible. The message is encrypted using the recipient's public key only and even the sender could not decrypt it.

However, in the comments you mention that you would like to minimize storage requirements. RSA would require e.g. 2048 bits for just the message. In comparison, with ECIES sending a one-time key with the message you are looking at a minimum of about 512 bits for a one AES block message, 256-bit curve and 128-bit MAC.

So it is not really wasteful to do it like CodesInChaos suggested.