The paper deals with the Markov process \(Z_ t(\omega)=(X_ t(\omega),Y_ t(\omega))\), where \(X_ t\in R\), \(Y_ t\in \{1,2\}\), whose transition probability is P(t,(x,i),d(y,i)), x,y\(\in R\), i,j\(\in \{1,2\}\). \(Z_ t\) corresponds to a particle moving along two parallel lines subject to random jumps from one line to another. The infinitesimal generator of this process has the form \[ \Omega =\left( \begin{matrix} \Omega_ 1+c_{11}(x)\\ c_{21}(x)\end{matrix} \begin{matrix} c_{12}(x)\\ \Omega_ 2+c_{\quad 22}(x)\end{matrix} \right) \] where \(\Omega_ i=(d/dx)a_ i(x)d/dx+b_ i(x)d/dx\), \[ \lim_{t\to 0}t^{-1}\{\int P(t,(x,i),d(y,i))-1\}=c_{ii}(x), \] \[ \lim_{t\to 0}t^{- 1}\int_{| y-x| \leq \epsilon}P(t,(x,i),d(y,j))=c_{ij}(x)\quad,\quad i,j=1,2. \] The authors study the classification of boundary points, minimal Markov process and sample path properties of \(Z_ t\). It is shown that \(Z_ t\) is conservative iff \(c_{11}(x)=-c_{12}(x)\), \(c_{21}(x)=-c_{22}(x)\) and the one-dimensional processes corresponding to \(\Omega_ i\) are conservative.