Null Strings Gauged and Reloaded, II: Consistent Classical Treatment of the Null Strings

We observed that the null strings, tensionless strings with Carrollian worldsheets, exhibit an extra gauge symmetry, \textit{Carroll-Weyl} gauge symmetry, which cannot be obtained from ultra-relativistic Carrollian limit of tensile strings. Due to the existence of this symmetry, the BMS$_3$ algebra of constraints, which is obtained as the Carrollian limit of two Virasoro algebras of the standard tensile strings, should be replaced with an BMS$_3$ algebra extended by a weight one operator. To establish further the existence and necessity of the Carroll-Weyl gauge symmetry, we carefully work through Hamiltonian analyses of constrained/gauged systems. We also discuss the extended BMS$_3$ algebra of constraints.