  In light of the observation of a relatively large $\theta_{13}$, the ever
popular Tri-Bimaximal (TBM) neutrino mixing which predicts a vanishing
$\theta_{13}$ needs modifications. In this paper, we shall discuss the
possibility of modifying it in a minimal way to fulfil this task. In the first
part, a neutrino mass matrix with three independent parameters, which leads to
the TM2 mixing, is obtained by analogy with that for the TBM mixing. In the
second part, a model that can realize the TM2 mixing is constructed with flavor
symmetries $A_4 \times U(1) \times Z_2 \times Z_2 \times Z_2$. It is the
variant of a model that gives the TBM mixing, with only one more flavon field
included. Furthermore, the imaginary vacuum expectation value (VEV) of this
flavon breaks the imposed CP symmetry and results in $\theta_{23}=45^\circ$ and
the maximal CP violation. This VEV is also helpful for providing a large enough
CP asymmetry required by the leptogenesis mechanism. Besides, this model
building approach can be generalized to the TM1 mixing in a straightforward
way.
