  In this paper, we formulate a supersymmetric extension of the
Gauss-Weingarten and Gauss-Codazzi equations for conformally parametrized
surfaces immersed in a Grassmann superspace. We perform this analysis using a
superspace-superfield formalism together with a supersymmetric version of a
moving frame on a surface. In constrast to the classical case, where we have
three Gauss-Codazzi equations, we obtain six such equations in the
supersymmetric case. We determine the Lie symmetry algebra of the classical
Gauss-Codazzi equations to be infinite-dimensional and perform a subalgebra
classification of the one-dimensional subalgebras of its largest
finite-dimensional subalgebra. We then compute a superalgebra of Lie point
symmetries of the supersymmetric Gauss-Codazzi equations and classify the
one-dimensional subalgebras of this superalgebra into conjugacy classes. We
then use the symmetry reduction method to find invariants, orbits and reduced
systems for two one-dimensional subalgebras in the classical case and three
one-dimensional subalgebras in the supersymmetric case. Through the solutions
of these reduced systems, we obtain explicit solutions and surfaces of the
classical and supersymmetric Gauss-Codazzi equations. We provide a geometrical
interpretation of the results.
