  For constants $\gamma \in (0,1)$ and $A\in (1,\infty)$, we prove existence
and uniqueness of a solution to the singular and path-dependent Riccati-type
ODE \begin{align*} \begin{cases} h'(y) = \frac{1+\gamma}{y}\big( \gamma -
h(y)\big)+h(y)\frac{\gamma + \big((A-\gamma)e^{\int_y^1
\frac{1-h(q)}{1-q}dq}-A\big)h(y)}{1-y},\quad y\in(0,1), h(0) = \gamma, \quad
h(1) = 1. \end{cases} \end{align*} As an application, we use the ODE solution
to prove existence of a Radner equilibrium with homogenous power-utility
investors in the limited participation model from Basak and Cuoco (1998).
