The Caughey-Dieness process, also known as the Brownian motion with two valued drift, is used in theoretical physics as an advanced model of the Brownian particle velocity if the resistant force is assumed to be dry friction. This process also appears in many other fields, such as applied physics, mechanics, astrophysics, and pure mathematics. In the present paper we are concerned with a more general process, skew Brownian motion with dry friction. The probability distribution of the process itself and of its occupation time on the positive half line are studied. The approach based on the Pugachev-Sveshnikov equation is used.