Asymmetric cipher with multiple public keys
Merely receiving a message and seeing that it has a valid signature does not provide any useful information. You also need to know who made the signature, or at least to have some information about them. If you don't know anything about who made the signature, then an adversary can generate their own message using whatever key they like.

In your scenario, there is apparently a group of trusted people. You want these people to be able to sign messages such that:


a recipient can verify that the message is a genuine message from a member of the group (as opposed to a message modified crafted by someone outside the group);
ordinary recipients do not know which member of the group produced a given message.


With just these conditions, you could make all senders use the same private key. However this would usually be a bad idea because it precludes many desirable properties, such as the ability for certain participants to tell who produced a given message, the ability to revoke the credentials of a participant who's defected, etc.

A common, relatively simple way to build such a system is to allow each participant to generate as many private keys as they like (possibly one for each message), and send the corresponding public keys to a certification authority (CA). The CA can choose what information it puts in the certificate. CAs for websites put the website name in the certificate, because that's what website certificates are for. A CA in your scenario would not put any identification in the certificates. The CA is thus an anonymity broker: it knows who is who but doesn't tell.

This scenario occurs with TPM: a TPM needs to be able to generate keys which cannot be tied to a specific TPM, but can be used to identify it with a particular service. What I described above is how multiple identities for TPM version 1 works. TPM 2 adds another method: direct anonymous attestations (DAA). I'm afraid that explaining the mathematics of DAA is beyond me. DAA may be a solution for you.