= Solution
The <metric tensor> is independent of $t$ and $\phi$, so
$$
\boxed{T=\partial_t,\qquad \Phi=\partial_\phi}
$$
are commuting <Killing vector fields>. For $0<a<M$ they span the continuous <Killing vector field> algebra; every constant linear combination is also a <Killing vector field>. They express <stationary spacetime> and axial <symmetry>, respectively. At the allowed endpoint $a=0$ the <spherical symmetry> of <Schwarzschild spacetime> enlarges the algebra: in addition to $T$, a basis of rotational <Killing vector fields> is
$$
J_x=-\sin\phi\,\partial_\theta-\cot\theta\cos\phi\,\partial_\phi,
\qquad
J_y=\cos\phi\,\partial_\theta-\cot\theta\sin\phi\,\partial_\phi,
\qquad
J_z=\partial_\phi.
$$
Their polar-coordinate expressions are singular at the axis, but the rotational fields are smooth there.
Let $U^a=dx^a/ds$ be the tangent to any affinely parametrized <geodesic>. For a <Killing vector field> $X$,
$$
\frac{d}{ds}(X_aU^a)
=U^aU^b\nabla_aX_b+X_bU^a\nabla_aU^b
=U^aU^b\nabla_{(a}X_{b)}=0.
$$
This is the <geodesic conserved quantity from a Killing vector>. With $p_a=g_{ab}U^b$, choose the <energy> and axial <angular momentum> conventions
$$
\boxed{E=-p_t=-g(T,U),\qquad L=p_\phi=g(\Phi,U).}
$$
For a massive particle with <proper time> parameter these are per unit rest mass; for a <null geodesic> their common scale depends on the <affine parameter> normalization. Expanding in the supplied <Boyer-Lindquist coordinates> gives
$$
\begin{aligned}
E&=\left(1-\frac{2Mr}{\Sigma}\right)\dot t
+\frac{2Mar\sin^2\theta}{\Sigma}\dot\phi,\\
L&=-\frac{2Mar\sin^2\theta}{\Sigma}\dot t
+\frac{\sin^2\theta}{\Sigma}
\left[(r^2+a^2)^2-a^2\Delta\sin^2\theta\right]\dot\phi.
\end{aligned}
$$
The factor of two between $g_{t\phi}$ and the coefficient of $dt\,d\phi$ is essential for these signs and coefficients.
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