  According to the no-hair theorem, astrophysical black holes are uniquely
characterized by their masses and spins and are described by the Kerr metric.
Several parametric spacetimes which deviate from the Kerr metric have been
proposed in order to test this theorem with observations of black holes in both
the electromagnetic and gravitational-wave spectra. Such metrics often contain
naked singularities or closed timelike curves in the vicinity of the compact
objects that can limit the applicability of the metrics to compact objects that
do not spin rapidly, and generally admit only two constants of motion. The
existence of a third constant, however, can facilitate the calculation of
observables, because the equations of motion can be written in first-order
form. In this paper, I design a Kerr-like black hole metric which is regular
everywhere outside of the event horizon, possesses three independent constants
of motion, and depends nonlinearly on four free functions that parameterize
potential deviations from the Kerr metric. This metric is generally not a
solution to the field equations of any particular gravity theory, but can be
mapped to known four-dimensional black hole solutions of modified theories of
gravity for suitable choices of the deviation functions. I derive expressions
for the energy, angular momentum, and epicyclic frequencies of a particle on a
circular equatorial orbit around the black hole and compute the location of the
innermost stable circular orbit. In addition, I write the metric in a
Kerr-Schild-like form, which allows for a straightforward implementation of
fully relativistic magnetohydrodynamic simulations of accretion flows in this
metric. The properties of this metric make it a well-suited spacetime for
strong-field tests of the no-hair theorem in the electromagnetic spectrum with
black holes of arbitrary spin.
