Summary: The focus of this work is a family of maps from the space of \(n\times n\) symmetric matrices, \(S^n\), into the space \(S^{\binom{n}{k}}\) for any \(k=1,\dots,n\), invariant under the conjugate action of the orthogonal group \(O^n\). This family, called the generated \(k\)-isotropic functions, generalizes known types of maps with a similar invariance property, such as the spectral, isotropic, primary matrix functions, multiplicative compound, and additive compound matrices on \(S^n\). We give necessary and sufficient conditions for the analyticity of these maps.