Which cryptographic key is contained in an X.509 certificate?

Symmetric keys are shared secret keys used in algorithms like AES and are never embedded in X.509 certificates, which are specifically designed for public-key infrastructure.

As defined in RFC 5280, an X.509 certificate contains the subject's public key in the SubjectPublicKeyInfo field, which is signed by a trusted Certificate Authority to bind the key to the subject's identity. This public key can then be used by others to encrypt data for the subject or to verify the subject's digital signatures. The corresponding private key is never included in the certificate and must remain exclusively under the subject's control.

The private key must remain secret and is never distributed via an X.509 certificate - including it would completely compromise the security model of PKI.

Asymmetric describes a category of cryptographic system, not a specific key type - X.509 certificates specifically carry only the public half of an asymmetric key pair.