SWT urged union officials to postpone strikes so that negotiations can take place, but union officials have indicated that they will not call off the strikes without a settlement in place.
Nine out of 10 trains were cancelled this morning in SWT's domain and into London's normally bustling Waterloo station as members of the RMT carried out the second day of a 48-hour strike over pay.
SWT today urged the union to postpone a second 48-hour strike planned for Monday and Tuesday to allow fresh talks to take place.
The union's acting general secretary, Vernon Hince, said last night that RMT officials were available to try to resolve the dispute, and he called on the company to return to the negotiating table.
The chairman of SWT, Graham Eccles, and Mr Hince agreed during a debate on BBC2's Newsnight programme last night to meet today to resume talks. However, the strike was not called off.
"We would meet this morning, no problem, but we are not going there to be told that we will only talk if the dispute is called off before we do," Mr Hince said.
The strike brought the company's 1,700 services across southern England to a near total halt. Next week's strike will cause further disruption as children will be returning to school after the Christmas break, increasing the number of rail users.
Special buses were again laid on for some season ticket holders in Surrey and Hampshire but many people decided to have a long weekend away from the office rather than try to struggle into work.
The union has rejected a 7.6% pay increase over two years and is holding out for the deal to cover 18 months, similar to an agreement with SWT drivers.
The union has also claimed that SWT has victimised its members following the disciplining of a union activist. But the company insisted that the demotion of Waterloo-based driver Greg Tucker was because of a safety related incident.
