We　investigate　the　singularity　formation　of　a　nonlinear　nonlocal　system.　This　nonlocal　system　is　a　simplified　one-dimensional　system　of　the　3D　model　that　was　recently　proposed　by　HOU　and　LEI　(Comm　Pure　Appl　Math　62(4):501-　564,　2009)　for　axisymmetric　3D　incompressible　Navier-Stokes　equations　with　swirl.　The　main　difference　between　the　3D　model　of　Hou　and　Lei　and　the　reformulated　3D　Navier-Stokes　equations　is　that　the　convection　term　is　neglected　in　the　3D　model.　In　the　nonlocal　system　we　consider　in　this　paper,　we　replace　the　Riesz　operator　in　the　3D　model　by　the　Hilbert　transform.　One　of　the　main　results　of　this　paper　is　that　we　prove　rigorously　the　finite　time　singularity　formation　of　the　nonlocal　system　for　a　large　class　of　smooth　initial　data　with　finite　energy.　We　also　prove　global　regularity　for　a　class　of　smooth　initial　data.　Numerical　results　will　be　presented　to　demonstrate　the　asymptotically　self-similar　blow-up　of　the　solution.　The　blowup　rate　of　the　self-similar　singularity　of　the　nonlocal　system　is　similar　to　that　of　the　3D　model.