Abstract
We give explicit expressions for canonical states labeling the vast majority of entanglement equivalent classes of symmetric states of qubits and efficient algorithms for reducing a given state to the representative of the class it belongs. This way, we achieve an almost complete classification under local unitary and local invertible transformations for symmetric states. The main tool is a technique introduced in this work, enabling to decompose in a unique way, spin symmetric states into a superposition of spin-$1/2$ coherent states. For the case of two qubits, the proposed decomposition reproduces the Schmidt decomposition and therefore, in the case of a higher number of qubits, can be considered as its generalization.