  In this paper, we consider the Wright's generalized Bessel kernel
$K^{(\alpha,\theta)}(x,y)$ defined by $$\theta
x^{\alpha}\int_0^1J_{\frac{\alpha+1}{\theta},\frac{1}{\theta}}(ux)J_{\alpha+1,\theta}((uy)^{\theta})u^\alpha\,\mathrm{d}
u, \qquad \alpha>-1, \qquad \theta>0,$$ where
$$J_{a,b}(x)=\sum_{j=0}^\infty\frac{(-x)^j}{j!\Gamma(a+bj)},\qquad
a\in\mathbb{C},\qquad b>-1,$$ is Wright's generalization of the Bessel
function. This non-symmetric kernel, which generalizes the classical Bessel
kernel (corresponding to $\theta=1$) in random matrix theory, is the hard edge
scaling limit of the correlation kernel for certain Muttalib-Borodin ensembles.
We show that, if $\theta$ is rational, i.e., $\theta=\frac{m}{n}$ with
$m,n\in\mathbb{N}$, $gcd(m,n)=1$, and $\alpha > m-1-\frac{m}{n}$, the Wright's
generalized Bessel kernel is integrable in the sense of
Its-Izergin-Korepin-Slavnov. We then come to the Fredholm determinant of this
kernel over the union of several scaled intervals, which can also be
interpreted as the gap probability (the probability of finding no particles) on
these intervals. The integrable structure allows us to obtain a system of
coupled partial differential equations associated with the corresponding
Fredholm determinant as well as a Hamiltonian interpretation. As a consequence,
we are able to represent the gap probability over a single interval $(0,s)$ in
terms of a solution of a system of nonlinear ordinary differential equations.
