Symmetries and their lie algebra of a variable coefficient Korteweg-de Vries hierarchy

Isospectral and non-isospectral hierarchies related to a variable coefficient Painleve integrable Korteweg-de Vries (KdV for short) equation are derived. The hierarchies share a formal recursion operator which is not a rigorous recursion operator and contains t explicitly. By the hereditary strong symmetry property of the formal recursion operator, the authors construct two sets of symmetries and their Lie algebra for the isospectral variable coefficient Korteweg-de Vries (vcKdV for short) hierarchy.