  We develop a systematic method to describe the moduli space of vacua of four
dimensional $\mathcal{N}=2$ class ${\cal S}$ theories including Coulomb branch,
Higgs branch and mixed branches. In particular, we determine the Higgs and
mixed branch roots, and the dimensions of the Coulomb and Higgs components of
mixed branches. They are derived by using generalized Hitchin's equations
obtained from twisted compactification of 5d maximal Super-Yang-Mills, with
local degrees of freedom at punctures given by (nilpotent) orbits. The crucial
thing is the holomorphic factorization of the Seiberg-Witten curve and
reduction of singularity at punctures. We illustrate our method by many
examples including ${\mathcal N}=2$ SQCD, $T_N$ theory and Argyres-Douglas
theories.
