Endogenous Growth: Balanced Growth Path with CRRA Utility
You have obtained

$$ \frac{c_{t+1}}{c_t}=[\beta(\alpha+1-\delta)]^{\frac{1}{\gamma}} \equiv 1+g$$

and

$$\frac{k_{t+1}}{k_t}=1+(1-\delta)-\frac{c_t}{k_t}$$

By equating you can show that there is a unique rule that maintains a balanced growth path

$$\frac{k_{t+1}}{k_t}=\frac{c_{t+1}}{c_t} \implies c_t =( 1-\delta-g)k_t$$ 

(too much consumption, by the way). This shows that the model has a unique balanced growth path. 

If you want further to argue that the economy will indeed choose this path, you have to invoke the Transversality condition (which constraints the consequences that a chosen path should have  on capital accumulation), and maybe the Inada condition that your chosen utility function satisfies.