I've been thinking about $SU(n)$invariant metrics on the odddimensional spheres $S^{2n1} \simeq SU(n)/SU(n1)$. For $S^1$, all such metrics are in correspondence with the positive reals. For $S^{3} \simeq SU(2)$, the tangent space is parallelizable, and so, such metrics are in correspondence with metrics on $R^2$. Does such a characterization exist for $S^5$, or higher.

Assume $n>2$. The $SU(n)$invariant metrics on the sphere $S^{2n1}$ are in bijection with $SU(n1)$ invariant metrics on $T_x S^{2n1}$ where $SU(n1)$ is realized as the stabilizer of some $x\in S^{2n1}$. This action is the sum of the defining $SU(n1)$module $V$ and the trivial real 1dimensional module $T$. The space of invariant real valued bilinear forms on this is 2dimensional and is spanned by the invariant Hermitian metric on the defining representation and some metric on the trivial representation. Notice that if $(\cdot,\cdot)$ is invariant and $v\in V, t\in T$, then $(v,t)=0$ since there is an $A\in SU(n1)$ such that $Av=v$. This is what makes the case $n>2$ different from the case $n=2$. 


Classification of $SU(n+1)$homogeneous metrics on $S^{2n+1}$:First, note that for any $n\geq1$, the round metric on the sphere $S^{2n+1}$ can be scaled by $t^2$ in the direction of the Hopf fibers $S^1\to S^{2n+1}\to \mathbb C P^n$, giving rise to a oneparameter family $g_t$ of $SU(n+1)$homogeneous metrics (so that $g_1$ is the original round metric).
In other words, they are always parameterized by (positive) real numbers. Alternatively, these $SU(n+1)$ homogeneous metrics on $S^{2n+1}$ are isometric to distance spheres in the complex projective space $\mathbb C P^{n+1}$. Both of the above statements follow from a more general result, namely W. Ziller's classification of all homogeneous metrics on spheres, see [Homogeneous Einstein metrics on spheres and projective spaces. Math. Ann. 259 (1982), no. 3, 351–358]. A few more details:An explicit formula for any $G$homogeneous metric on the homogeneous space $G/H$, in terms of an $Ad(H)$invariant decomposition $g= h\oplus p$ of the Lie algebra of $G$ is: $$\langle ,\rangle=h_{p_0} + \sum_{i=1}^r\alpha_i B_{p_i},$$ where $p= p_0\oplus\dots\oplus p_r$ is a decomposition so that the $H$ representations on $p_i$ are not equivalent, $H$ acts trivially on $p_0$ and irreducibly on $p_i$, $i=1,\dots r$; $h_{p_0}$ is any inner product on $p_0$; $B$ is a biinvariant metric on $G$ and $\alpha_i>0$ are real parameters. Any $G$homogeneous metric is defined by choosing these parameters. In the case of $SU(n+1)$, this decomposition is $p=p_0\oplus p_1$ and $\dim p_0=1$, so (up to renormalization) there is only one parameter, $\alpha_1$ to be chosen (that I called $t$ above). EDIT. Since I claim the metrics are parameterized by one positive real number (and another answer above claims there must be two real parameters), a clarification is in order here. The point is that, indeed, there are two parameters ($h_{p_0}$ and $\alpha_1$ in the notation above), nevertheless it is always possible to divide the entire metric by the first one (the number that determines the metric $h$ on the 1dim space $p_0$), which leaves us with just one parameter. Treating the family as having the $2$ parameters is slightly ambiguous because lots of metrics will be simple rescaling of other ones. In my description above, they are pairwise nonconformal. 

