There are only two places in the Bible that refer to the “law of Christ.” The first is in 1 Corinthians 9:21 (ἔννομος Χριστοῦ) as Paul clarifies that he does not abandon righteous living to reach the unrighteous with a message of righteousness. He doesn’t abandon obedience to Christ when calling others to obey Christ.
The second time the phrase is used in Galatians 6:2 (τὸν νόμον τοῦ Χριστοῦ), though Galatians was probably written five to six years before 1 Corinthians. To the Galatians Paul explains at least one way to live out the law of Christ.
Bear one another's burdens, and so fulfill the law of Christ.
This is called the law of Christ because this is Christ’s custom, His rule, His norm. The Son of God picks up another man’s load and carries it for him. The word for “burdens” applies to something “particularly oppressive” or something that “proves exhausting” (BDAG). The plural form indicates at least a diversity of possible burdens if not a multiplicity of burdens. It might be a big one, it might be more than one.
In the context of Galatians 6 the burden could be a brother caught in a transgression. It could also be a brother caught in an affliction. There are all sorts of ways that we struggle and suffer. Remember, “A friend loves at all times, and a brother is born for adversity” (Proverbs 17:17). It is certainly easier not to bear another burden.
This is a communion encouragement, not an exhortation. As a church you do well in bearing the burdens of one another and so fulfilling the law of Christ. Yes, excel still more, as you feed on Christ’s flesh and follow His example of death that brings life (2 Corinthians 4:12).